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ELEMENTS  OF  SYNTHETIC   SOLID   GEOIETEY. 


^><^m 


ELEMENTS 


SYNTHETIC  SOLID   GEOMETRY 


^ 


BY 


N.  F.  DUPUIS,   M.A.,  F.R.S.C. 

PBOyBSSOR  OF  PCBK  MATHEMATICS   IN  THE  UNIVERSITY  OP  QuEEN'S    COLLEGE, 

Kingston,  Canada 


MACMILLAN   AND   CO. 

AND    LONDON 
1893 

All  rights  reserved 

3  '7  0<?7 


Copyright,  1893, 
By  MACMILLAN  AND  CO, 


J  S.  Gushing  &  Co.  — Berwick  &  Smith. 
Boston,  Mass.,  U.S.A. 


Y  '^     (  Library 


PREFACE. 


The  matter  of  the  present  work  has,  with  some  varia- 
tions, been  in  manuscript  for  a  number  of  years,  and  has 
formed  the  subject  of  an  annual  course  of  lectures  to 
mathematical  students  by  whom  the  subject  has  been 
well  received  as  one  of  the  most  interesting  in  the  earlier 
part  of  a  mathematical  course. 

I  have  been  induced  to  present  the  work  to  the  public, 
partly,  by  receiving  from  a  number  of  Educationists 
inquiries  as  to  what  work  on  Solid  Geometry  I  would 
recommend  as  a  sequel  to  my  Plane  Geometry,  and  partly, 
from  the  high  estimate  that  I  have  formed  of  the  value 
of  the  study  of  synthetic  solid  geometry  as  a  means  of 
mental  discipline. 

To  me  it  seems  to  exercise  not  only  the  purely  intel- 
lectual powers  in  the  development  of  its  theorems,  but 
also  the  imagination  in  the  mental  building-up  of  the 
necessary  spatial  figures,  and  the  eye  and  the  hand  in 
their  representations. 

In  this  work  the  subject  is  carried  somewhat  farther 
than  is  customary  in  those  works  in  which  the  subject 
of  solid  geometry  is  appended  to  that  of  plane  geometry. 


VI  PREFACE. 

but  the  extensions  thus  made  are  fairly  within  the  scope 
of  an  elementary  w^rk,  and  are  highly  interesting  and 
important  in  themselves  as  forming  valuable  aids  to  the 
right  understanding  of  the  more  transcendental  methods. 

It  appears  to  me  that  it  is  a  prevalent  custom  to  lay 
too  little  stress  on  synthetic  methods  as  soon  as  plane 
geometry  is  passed,  and  to  hurry  the  student  too  rapidly 
into  the  analytic  methods.  If  mathematical  knowledge 
is  all  that  is  required,  this  may  possibly  be  an  advan- 
tageous course ;  but  if  mental  culture  is,  as  it  should 
be,  the  chief  end  in  a  university  education,  this  custom- 
ary usage  is  not  the  best  one. 

I  have  found  it  convenient  to  divide  the  work  into 
four  parts,  each  of  which  is  further  divided  into  sec- 
tions. 

The  first  part  deals  with  a  consideration  of  the  descrip- 
tive properties  of  lines  and  planes  in  space,  of  the  poly- 
hedra,  and  of  the  cone,  the  cylinder,  and  the  sphere. 

Here  I  would  feel  like  apologising  for  the  introduction 
of  a  new  term,  were  it  not  that  I  believe  that  its  intro- 
duction will  be  fully  justified  by  a  careful  perusal  of 
the  work. 

Legendre,  in  his  notes  to  his  geometry,  proposed  to 
use  the  word  '  corner '  (coin)  for  the  figure  formed  by 
the  meeting  of  two  planes,  and  he  considered  that  the 
different  polyhedral  angles  should  receive  special  names 
as  being  geometrical  figures  of  different  species.   Without 


PREFACE.  vii 

discussing  this  idea,  I  have  employed  the  word  'corner' 
to  denote  a  solid  or  polyhedral  angle  of  not  less  than 
three  faces,  while  I  have  retained  the  expression  'dihedral 
angle'  in  its  usual  sense.  If  a  dihedral  angle  be  cut  by 
a  plane,  this  cutting  plane  necessarily  cuts  through  both 
faces,  and  the  figure  of  intersection  is  a  plane  angle. 
Whereas,  if  any  polyhedral  angle  be  cut  by  a  plane 
which  intersects  all  its  faces,  the  figure  of  section  is  not 
a  simple  angle,  but  a  polygon.  Thus  the  plane  angle  and 
the  dihedral  have  this  in  common,  that  they  can  both 
be  measured  by  the  same  kind  of  angular  unit,  while 
the  affinities  of  the  polyhedral  angle  are  with  the 
polygon. 

Moreover,  the  trihedral  angle  is  a  geometrical  func- 
tion of  three  plane  angles  and  three  dihedral  angles, 
neither  of  which  exists  without  the  other,  and  every 
polyhedral  angle  is  a  geometrical  function  or  combina- 
tion of  plane  and  dihedral  angles,  and  these  form  its 
elements.  Hence  I  have  used  the  term  'three-faced 
corner'  for  'trihedral  angle,'  and  generally  'w-faced 
corner '  for  '  7i-hedral  angle.'  This  nomenclature  is  very 
convenient ;  but  if  any  Teacher  prefers  the  older  forms, 
he  can  readily  make  the  necessary  change  in  language. 

The  rectangular  parallelepiped  should  certainly  be 
supplied  with  some  convenient  name.  I  have  adopted 
the  term  '  cuboid,'  as  proposed  by  Mr.  Hay  ward,  as  being 
both  convenient  and  suggestive. 


VIU  PREFACE. 

The  second  part  of  the  work  deals  with  areal  rela- 
tions, that  is,  the  relations  among  the  areas  of  squares 
and  rectangles  on  characteristic  line-segments  of  the 
prominent  spatial  figures. 

The  majority  of  the  results,  besides  being  highly  inter- 
esting in  themselves,  form  data  for  subsequent  higher 
work. 

The  third  part  is  devoted  to  stereometry  and  planimetry. 
In  this  are  developed  the  principal  rules  and  formulae 
for  the  measurement  of  volumes  and  surfaces  of  the 
more  prominent  spatial  figures  which  admit  of  such 
measurement,  and  a  special  section  is  given  to  the  con- 
sideration of  volumes  and  surfaces  generated  by  moving 
areas  and  lines,  and  to  the  development  of  the  theorems 
of  Pappus  or  Guldinus. 

The  fourth  and  last  part  begins  with  an  explanation. 
of  the  principles  of  conical  or  perspective  projection. 
By  the  application  of  these  principles  in  projecting  a 
circle  into  a  cone  and  cutting  the  cone  by  a  plane,  the 
student  is  introduced  to  the  conic,  and  is  led  to  under- 
stand its  meaning,  and  the  relations  of  the  various  conies 
to  one  another. 

The  more  common  properties  of  the  conies  are  then 
easily  obtained  through  a  study  of  the  curve  as  a  plane 
section  of  a  circular  cone.  The  latter  half  of  this  part 
is  given  to  spheric  geometry.  The  spheric  figure  (tri- 
angle and  polygon)   is  considered  as   the  section  of  a 


PREFACE.  IX 

corner  by  a  sphere  whose  centre  is  at  the  apex  of  the 
corner.  The  study  of  spheric  figures  is  thus  brought 
into  line  with  the  study  of  the  corner  or  solid  angle, 
and  the  leading  properties  of  the  spheric  triangle  are 
thus  most  easily  and  directly  obtained. 

The  whole  work  is  presented  to  the  younger  mathe- 
matical reader  in  the  hope  that  it  may  prove  worthy  of 
his  careful  attention. 

At  the  close  of  the  work  there  is  a  large  collection  of 
miscellaneous  exercises,  many  of  Avhich,  being  connected 
with  the  subjects  of  inversion  and  of  polar  reciproca- 
tion in  space,  are  highly  suggestive. 

I  have  to  acknowledge  my  indebtedness  to  Mr.  W.  R. 
Sills  for  assistance  in  reading  the  proof-sheets. 

N.  F.  D. 

Queen's  College, 
Oct.  1,  1893. 


CONTENTS. 


PAET  I. 

DESCRIPTIVE   GEOMETRY. 
Sect.  Paob 

1.  The  Line  and  the  Plane.     Intersection.    Axial  Pencil. 

Generation  of  Plane.     Normal.     Cone-circle.     Con- 
structions      .        4 

2.  Two  Planes.     Dihedral  Angle.     Planar  and  Non-planar 

Figures.     Skew  Quadrilateral 17 

3.  Sheaf  of  Lines  or  Planes.      Solid  Angle  or  Comer. 

Properties  of  Three-faced  Comers  or  Trihedral  An- 
gles.    Certain  Loci 29 

4.  Polyhedra.     Euler's  Theorem  (?+  F=E  +  2.    Tetra- 

hedron.   Parallelepiped.    Pyramid.    Prism.     Regular 
Polyhedra.     Constructions.    Nets 45 

5.  Cone,  Cylinder,  and  Sphere.    Tangent  Lines  and  Planes. 

Conditioned  Spheres 61 

PA^T  II. 

AREAL  RELATIONS   INVOLVING  LINE-SEGMENTS  OF 
SPATIAL  FIGURES. 

1.  Tetrahedron.     Parallelepiped.-   Cuboid.    Regular  Poly- 

hedra       78 

2.  Sphere.     Radical  Plane.    Radical  Line.    Tangent  Cone. 

Polar       93 


Xll  CONTENTS. 


PAKT  III. 

STEREOMETRY   AND   PLANIMETRY. 

Sect.  Page 

1.  Polyhedra.     Theory  of  Laminae.     Frustum.     Prisraa- 

toid.     Prismoidal  Formula.     Applications     ....     100 

2.  Closed  Cone.    Frustum.    Cylinder.    Sphere.    Zone  and 

Segment 124 

3.  Special  Processes.     A.    Spatial  Figures  generated  by 

Translation  of  a  Plane  Figure.  Pyramid.  Cone. 
Sphere.  Groin,  etc.  B.  Figures  of  Revolution. 
C.  Mean  Centre  of  Area.  Theorems.  Guldinus' 
Theorem  for  Volumes 132 

4.  Areas    of    Surfaces.      Developable    Surfaces.       Cone. 

Cylinder.  Sphere.  Mean  Centre  of  Figure.  Guldi- 
nus' Theorem  for  Surfaces 155 


PAKT  IV. 
PROJECTIONS  AND   SECTIONS. 

1.  Perspective  Projection.   Various  Projections.  Vanishing 

Point  and  Line.  Anharmonic  Relations.  General 
Theorem 165 

2.  Plane  Sections  of  the  Cone.     Classification  of  Conies. 

Degraded  Forms.  Common  Properties.  Foci  and 
Directrices.  Special  Study  of  Ellipse.  Theorems  of 
Apollonius.    The  Parabola 175 

3.  Spheric  Sections.     General  Ideas  of  Spheric  Geometry. 

Spheric  Line.  Pole  and  Equator.  Lune.  Spheric 
Triangle.  Polar  Triangle.  Properties  of  Spheric 
Triangles.  Spherical  Excess.  Supei-posability  and 
Symmetry.    Cases  of  Ambiguity 197 


SOLID  OE  SPATIAL  GEOMETET. 
Part  I. 

DESCRIPTIVE   GEOMETRY. 

1.  Solid  or  Spatial  Geometry,  or  the  Geometry  of 
Si^ace,  deals  with  the  properties  and  relations  of  figures 
not  confined  to  one  plane  (P.  Art.  19).^ 

The  elements  of  spatial  figures  are  the  point,  the  line, 
the  curve,  the  plane,  and  the  curved  surface.  The  first 
four  of  these  are  defined  in  plane  geometry  (P.  Arts.  12, 
14,  17) ;  but  we  repeat  here  the  definition  of  the  plane, 
as  upon  that  definition  several  corollaries  and  other 
definitions  depend. 

Def.  K  plane  is  a  surface  such  that  the  join  of  any 
two  arbitrary  points  in  it  lies  wholly  in  the  surface  and 
coincides  with  it. 

Cor.  1.  A  line  cannot  lie  partly  within  a  plane  and 
partly  without  it.  For  the  part  within  the  plane  must 
have  at  least  two  points  in  the  plane,  and  must  there- 
fore coincide  with  the  plane  throughout  its  whole  extent. 

1  References  marked  Rare  to  the  Author's  '  Geometry  of  the  point, 
line,  and  circle  in  the  plane.' 

1 


2  SOLID   OR   SPATIAL   GEOMETRY. 

Cor.  2.  A  line  not  coincident  with  a  given  plane 
meets  the  plane  at  only  one  point. 

2.  A  plane  is  not  necessarily  limited  in  extent ;  or,  in 
other  words,  a  plane  extends  to  infinity  in  all  its  direc- 
tions. For  the  plane  must  be  coextensive  with  every 
coincident  line. 

Every  plane  thus  theoretically  divides  all  space  into 
two  parts,  one  lying  upon  each  side  of  the  plane.  The 
use  of  planes  thus  considered  is  common  in  spherical 
astronomy. 

3.  In  plane  geometry  the  geometric  figure  is  drawn 
upon  the  plane  of  the  paper,  which  properly  represents 
the  plane  upon  which  the  figure  is  supposed  to  lie.  In 
spatial  geometry,  however,  we  have  only  one  plane,  that 
of  the  paper,  to  stand  for  and  represent  all  the  planes 
which  may  be  involved  in  any  spatial  figure.  This  is 
an  unavoidable  source  of  confusion  to  beginners,  as  the 
pictured  figures  in  spatial  geometry  are  not  representa- 
tions of  the  real  figures  in  the  same  sense  as  in  plane 
geometry. 

Thus  equal  line-segments  and  equal  angles  in  a  spatial 
figure  will  not,  in  general,  appear  as  equal  segments  or 
equal  angles  in  the  pictured  representation.  So,  also, 
squares  and  circles  in  space  will  not,  in  general,  appear 
as  squares  and  circles  on  our  single  available  plane,  that 
of  the  paper.  Properly  constructed  models  simplify 
matters  to  a  very  great  extent,  and  should  be  employed 
whenever  available.  The  construction  of  proper  models 
is,  however,  always  difficult,  and  often  impracticable, 
and  for  several  reasons  they  cannot  serve  all  the  pur- 
poses of  a  diagram.     And  hence   beginners  should   ac- 


DESCRIPTIVE   GEOMETRY.  3 

custom  themselves  to  reading  and  interpreting  spatial 
diagrams.  These  diagrams  can  be  considered  only  as 
an  aid  to  building  up  the  figure  in  the  imagination,  and 
facility  in  reasoning  from  such  diagrams  will  depend 
very  largely  upon  the  readiness  with  which  the  reasoner 
can  make  this  imaginary  construction.  The  student  is 
accordingly  advised  to  give  some  care  and  patience  to 
the  constructing  of  spatial  diagrams. 

To  represent  a  plane  we  usually  represent  a  rectangular 
segment  of  the  plane,  and  this  generally  appears  in  the 
diagram  as  some  form  of  parallelogram. 


SECTION  1. 
The  Line  and  the  Plane. 

4.  Theorem.  Two  planes  which  coincide  in  part  coin- 
cide altogether. 

Proof.  The  part  throughout  which  the  planes  coincide 
must  be  part  of  a  plane,  and  must  therefore  admit  of  an 
indefinite  number  of  arbitrary  points  being  taken  within 
it,  of  which  no  three  are  in  line.  These  points  taken 
two  and  two  determine  an  indefinite  number  of  arbitrary- 
lines  which  coincide  in  part  with  both  planes.  And  the 
planes  thus  coinciding  (Art.  1.  Cor.  1)  along  an  indefinite 
number  of  arbitrary  lines,  coincide  altogether,  and  form 
virtually  but  one  plane. 

Def.  An  indefinite  number  of  lines  can  lie  in  one  plane. 
The  totality  of  these  is  called  Biplane  of  lines,  although 
the  lines,  having  only  one  dimension,  do  not  make  up 
any  portion  of  the  plane  in  which  they  lie. 

5.  Theorem.  The  figure  of  intersection  of  two  planes 
is  a  line. 

Proof.  Let  U  and  V  be  two 
planes,  and  let  A  and  B  be  any 
two  points  in  their  figure  of  in- 
tersection. Join  ^,  ^  by  a  line. 
Then,  since  A  and  B  are  two 
points  in  U,  the  join  AB  lies  wholly  in  U  (Art.  1.  Del). 

4 


THE  LINE  AND   THE  PLANE.  5 

For  a  similar  reason  the  join  AB  lies  wholly  in  V. 
Hence  it  is  common  to  the  planes,  and  is  their  figure  of 
intersection;  and  thus  the  figure  of  intersection  is  a 
line. 

Cor.  1.  Any  number  of  planes  may  have  one  com- 
mon line.  For  if  they  pass  through  the  same  two  points, 
A  and  B,  they  have  the  join  of  A  and  .B  as  a  common 
line. 

Def.  A  group  of  planes  having  one  common  line  is 
an  axial  pencil,  and  the  line  is  the  axis.  In  contra- 
distinction to  this  the  pencil  of  lines  in  a  common  plane 
(P.  Art.  203.  Def.)  is  called  a,  flat  pencil. 

Cor.  2.  As  the  line  of  section  of  two  planes  cannot 
return  into  itself  and  form  a  closed  plane  figure,  so  two 
planes  cannot  form  a  closed  spatial  figure. 

6.  Theorem.     Through  any  three  points  not  in  line, 

1.  One  plane  can  pass. 

2.  Only  one  plane  can  pass. 
A,  B,  C  are  any  three  points  not  in  line. 

1.  One  plane  can  pass  through 
A,  B,  and  C. 

Proof.  Let  the  plane  contain- 
ing A  and  B  be  rotated  about 
the  join  of  A  and  B. 

In  a  complete  revolution  this  plane  passes  through 
every  point  in  space,  and  therefore  in  some  position,  U, 
it  passes  through  C 


6  SOLID   OR   SPATIAL  GEOMETRY. 

2.   Only  one  plane  can  pass  through  A,  B,  and  C 

Proof.  Take  D,  E,  any  points  in  the  joins  AC  and 
BC  respectively.  Then  D  and  E  and  their  join  lie  in 
U,  and  in  every  plane  through  A,  B,  and  C.  Therefore 
every  plane  through  A,  B,  and  C  coincides  with  U,  and 
forms  with  U  virtually  but  one  plane. 

Cor.  1.  Any  three  points  not  in  line  determine  a 
single  plane. 

Def.  Any  number  of  elements  so  disposed  as  to  lie 
in  one  and  the  same  plane  are  said  to  be  complanar  or 
coplanar.  Thus  all  the  parts  of  a  figure  in  plane  geom- 
etry are  complanar. 

Cor.  2.  Two  intersecting  lines  are  complanar  and 
determine  one  plane. 

For,  taking  a  point  in  each  line,  and  the  point  of  inter- 
section, we  have  three  points  not  in  line,  and  the  plane 
through  these  is  the  plane  of  the  lines. 

Cor.  3.     Parallel  lines  are  complanar. 

For  they  have  a  common  point  at  infinity  (P.  Art. 
220.  Def.). 

7.  Generation  of  a  plane. 

L  and  Jf  are  any  two  lines 
intersecting  in  C,  and  N  is 
a  third  line  intersecting  L 
in  B,  and  M  in  A.  Then 
L,  M,  ^are  complanar. 

1.  When  L  and  M  are  fixed  and  N  is  variable,  N 
generates  a  plane. 

Therefore,  a  plane  is  generated  by  a  variable  line 
which  is  guided  by  two  intersecting  fixed  lines. 


THE   LINE   AND   THE  PLANE. 


Def.  The  variable  line  N  is  called  the  generator,  and 
the  fixed  guiding  lines  are  directors. 

2.  Let  C  go  to  infinity,  and  L  and  M  become  parallel. 
Therefore,  a  plane   is   generated  by  a  variable   line 

guided  by  two  fixed  parallel  lines. 

3.  Let  the  point  A  remain  fixed,  while  B  moves 
along  L. 

Then,  a  plane  is  generated  by  a  variable  line  which 
passes  through  a  fixed  point  and  is  guided  by  a  fixed  line. 

4.  Let  the  point  A  go  to  infinity ;  i.e.  let  the  genera- 
tor N,  fixed  in  direction  only,  be  guided  by  the  fixed 
line  L. 

Then,  a  plane  is  generated  by  a  variable  line  having 
a  fixed  direction  and  guided  by  a  fixed  line. 

8.  Theorem.  At  the  point  of  intersection  of  any  two 
lines  a  third  line  can  be  perpendicular  to  both. 

AB  and  CD  are  lines  intersecting  in  0.     Then  some 
line    OP   is    perpendicular    to 
both  AB  and  CD. 

Proof.  Let  OP  be  ±  to  AB, 
and  let  it  revolve  about  AB  as 
an  axis,  O  being  fixed,  until  it 
comes  into  the  plane  of  AB  and 
CD  at  OE  and  at  OF.  Then 
AB,  CD,  EF  are  complanar. 

ZAOE  =  AAOP=-[; 

.-.  Zi)OjEis<al. 

Similarly  Z  BOF  =  Z  BOP  =  1 ; 

and  .-.  /.DOFi^>a.~\. 


(hyp.) 


8  SOLID   OR   SPATIAL  GEOMETRY. 

Therefore,  in  revolving  OP  from  the  position  OE  to 
the  position  OF  the  Z  DOP  changes  from  less  than  a 
right  angle  at  DOE  to  greater  than  a  right  angle  at  DOF; 
and  hence  at  some  intermediate  position  OP  is  _L  OD. 

Cor.  If  AB  is  ±  to  CD,  and  OP  is  ±  to  both, 
we  have  three  lines  mutually  perpendicular  to  each 
other. 

Def.  1.  Three  concurrent  lines  mutually  perpendicu- 
lar to  one  another  are  called  the  three  rectangular  axes 
of  space,  and  their  planes  are  the  rectangular  co-ordi- 
nate planes  of  space.  These  three  lines  admit  of  length 
measures  in  three  directions,  each  perpendicular  to  the 
other  two.  Hence,  space  is  said  to  be  of  three  dimen- 
sions, or  to  contain  three  dimensions,  and  it  is  frequently 
spoken  of  as  tri-dimensional  space,  in  contradistinction 
to  the  two-dimensional  space  of  a  single  plane,  or  of 
plane  geometry. 

Def.  2.  A  line  lying  in  a  particular  plane  is  a  planar 
line  of  that  plane ;  and  when  only  one  plane  is  under 
consideration,  a  planar  line  will  mean  a  line  in  that 
plane. 

Def.  3.  When  OP  is  perpendicular  to  both  AB  and 
CD,  it  is  perpendicular  to  the  plane  which  these  lines 
determine  (Art.  6.  Cor.  2). 

OP  is  then  a  normal  to  the  plane,  and  0  is  the  foot 
of  the  normal. 

Also,  the  plane  is  a  normal  plane  to  the  line  OP. 

9.  Theorem.  A  normal  to  a  plane  is  perpendicular  to 
every  planar  line  through  the  foot  of  the  normal. 


THE  LINE  AND  THE  PLANE. 


9 


OP  is  ±  to  OA  and  OB,  and  OC  is  any  line  through  0 
complanar  with  OA  and  OB.    Then 
OP  is  ±  to  OC. 

Proof.  Take  0^=  0B=  any  con- 
venient length.  Join  AB,  cutting 
OC  in  C,  and  join  PA,  PB,  PC. 

The  right-angled  triangles  POA 
and  POB  are  congruent,  and  there- 
fore PA  =  P5.    Hence  the  A  ^P^ 

and  u405  are  each  isosceles,  and  PC  and  OC  are  lines 
from  the  vertices  to  the  common  base  AB. 

...  PB^  _  pC^  =  5(7 .  CM  =  OPT-  -  OC;     (P.  Art.  174.) 
and  .  •.  P:^-  -  OB'  =  PC  -  0C\ 

But  POB  being  a  "I,  (hyp.) 

PB?  -0B?=  OP""  =  PC  -  OC. 
.'.  Z  POC  is  a  1. 

Cor.  1.  If  0  is  fixed  while  OA  revolves  about  OP  as 
an  axis,  OA  generates  a  plane  to  which  OP  is  a  normal. 

Def.  A  line  is  perpendicular  to  a  line  which  it  does 
not  meet  when  a  plane  containing  one  of  the  lines  can 
have  the  other  as  a  normal. 

Cor.  2.  A  normal  to  a  plane  is  perpendicular  to  every 
line  in  the  plane,  and  all  normals  to  the  same  plane  are 
parallel  to  one  another. 

Cor.  3.  From  any  point  without  or  within  a  plane,  only 
one  normal  can  be  drawn  to  the  plane. 

10.  Theorem.  Of  the  line-segments  from  a  point  with- 
out a  plane  to  the  plane :  — 

1.  The  shortest  is  along  the  normal  through  the  point. 


10 


SOLID   OR   SPATIAL    GEOMETRY. 


2.  The  feet  of  equal  segments  are  equally  distant  from 
the  foot  of  the  normal,  and  conversely. 

3.  Of  unequal  segments, 
the  longer  lies  further  from 
the  normal  than  the  shorter 
does,  and  conversely. 

P  is  any  point,  and  PO  is 
normal  to  the  plane  U,  not 
passing  through  P.  A,  B,  C 
are  points  in  U. 

1.  PO  is  <  PA,  A  being  any  point  in  U  other  than  0. 

Proof.  A  POA  is  a  1 ;  (Art.  9.  Cor.  2.) 

.-.  Z  P^O  is  acute,  and  PO<PA;  (P.  Art.  62.) 
and  the  normal  segment  PO  is  the  shortest  segment  from 
P  to  the  plane  U. 

2.  PA  =  PB ;  then  OA  =  OB. 

Proof.  The  right-angled  triangles  POA  and  POB  have 
their  hypothenuses  equal,  and  the  side  PO  in  common. 
They  are  therefore  congruent  (P.  Art.  65),  and  OA  =  OB. 

Conversely,  if  OA  =  OB,  the  congruence  of  the  same 
triangles  gives  PA  =  PB. 

3.  PC  is  >PA;  then  OC  is  >  OA. 

For  the  two  triangles  POA,  POC,  being  each  right- 
angled,  give 

PC^  =  PO^-^OC''', 

and  PA^  =  PO^+OA^; 

...  PC'  _  PA^  =  OC'  -  OAK 
But  PC>PA;   ..OC>OA. 

And  conversely,  if  0C>  OA,  then  PC>PA. 


THE  LINE  AND  THE  PLANE.  11 

Cor.  When  PA  =  PB,  OA  =  OB.  Therefore,  if  PA 
is  of  constant  length  and  variable  in  position,  the  foot  A 
describes  a  circle  having  0  as  centre  and  OA  as  radius. 
The  generation  of  this  circle  from  a  fixed  point,  P,  by  a 
line  segment,  PA,  of  constant  length,  is  similar  to  that 
of  the  circle  in  plane  geometry  (P.  Art.  92),  except  that 
in  the  present  case  the  fixed  point  is  not  in  the  plane  of 
the  circle. 

Def.  1.  The  circle  described  on  [/"with  the  vector  PA, 
and  from  the  fixed  point  P,  has  a  relation  to  the  cone,  to 
be  considered  hereafter,  and  we  shall  accordingly  call  it 
a  cone  circle  to  the  vertex  P. 

Evidently  any  circle  may  be  considered  as  a  cone 
circle,  and  when  so  considered,  it  has  an  indefinite  num- 
ber of  vertices,  all  lying  upon  the  line  which  passes 
through  its  centre  and  is  normal  to  its  plane. 

Def.  2.  The  distance  of  a  point  from  a  plane  is  the 
length  of  normal  intercepted  between  the  point  and  the 
plane. 

11.  Def.  1.  The  projection  of  a  point  on  a  plane  is 
the  foot  of  the  normal  from  the  point  to  the  plane,  and 
the  projection  of  a  line-segment  on  a  plane  is  the  join 
of  the  projections  of  its  end-points  upon  the  plane. 

It  follows,  then,  that  the  projection  of  a  line  upon  a 
plane  which  it  meets  is  the  planar  line  which  passes 
through  the  point  where  the  given  line  meets  the  plane, 
and  through  the  foot  of  the  normal,  drawn  from  any  point 
on  the  given  line  to  the  plane. 

Def.  2.  The  angle  between  a  given  line  and  its  pro- 
jection upon  a  plane  is  taken  to  be  the  angle  between 
the  given  line  and  the  plane. 


12  SOLID   OR   SPATIAL   GEOMETRY. 

Def.  3.  The  angle  between  two  non-complanar  lines 
is  the  angle  between  two  intersecting  lines  respectively- 
parallel  to  the  given  lines. 

12.  Theorem.  The  angle  between  a  line  and  its  pro- 
jection on  a  plane  is  less  than  the  angle  between  the 
given  line  and  any  planar 
line  not  parallel  to  the  pro- 
jection. 

The  line  PO  meets  the 
plane  U"  in  0;  ON  is  the 
projection  of  OP  on  U;  OA 
is  a  line  through  0,  parallel 
to  the  planar  line  L,  which  is  not  parallel  to  the  pro- 
jection ON. 

Then  Z  PON  is  <  Z  POA. 

Proof.     From  P  draw  PN  perpendicular  to  ON.     PN 
is  normal  to  the  plane  U  (Art.  11.  Def.  1). 
Take  OA  =  ON  and  join  PA  and  AN 
Since  Z  PNA  =  1,  PA  is  >  PN. 

And  in  the  triangles  POA  and  PON,  PO  is  common, 
OA  =  ON,  and  PA  >  PN; 
.:  Z  POA  is  >  Z  PON.  (P.  Art.  67.) 

And  as  L  is  any  planar  line  not  parallel  to  ON,  the 
Z  PON,  between  PO  and  its  projection  on  U,  is  less 
than  that  between  PO  and  any  line  in  the  plane,  not 
parallel  to  ON 

Cor.  1.  Since  two  intersecting  lines  make  with  one 
another  two  angles  which  are  supplementary  (P.  Art. 


SPATIAL  CONSTRUCTION.  13 

39),  we  may  say  more  accurately  that  the  angles,  between 
a  line  and  its  projection  upon  a  plane,  are  the  least  and 
the  greatest  of  all  the  angles  made  by  the  given  line 
with  lines  lying  in  the  plane. 

Cor.  2.  Since  0,  P,  N  are  complanar  (Art.  6),  and 
/.  PNO  is  a  1,  the  Z  OPN  is  the  complement  of  the 
Z  PON.  Therefore  the  angle  between  a  line  and  a  plane 
is  the  complement  of  the  angle  between  the  line  and  a 
normal  to  the  plane. 

Cor.  3.     Let  OB  be  a  planar  line  _L  to  OP. 

Since  PN  is  normal  to  U,  OB  is  i.  to  PN  (Art.  9. 
Cor.  2) ;  and  hence  OB,  being  ±  to  OP  and  PN,  is  ±  to  ON. 

Therefore  planar  lines  which  are  perpendicular  to  any 
line  that  meets  their  plane  are  also  perpendicular  to  the 
projection  of  that  line  upon  the  plane. 

13.  Def.  A  line  is  parallel  to  a  plane  when  it  meets 
that  plane  at  infinity. 

Cor.  Any  plane  through  one  of  two  parallel  lines  is 
^  parallel  to  the  other  line. 

For  if  L  and  M  be  two  parallel  lines,  and  the  plane  U 
contains  L  and  not  M,  it  can  meet  Jf  only  where  L  meets 
M.  But  L  and  Jf  meet  at  infinity  (P.  Art.  220)  ;  there- 
fore M  meets  U  at  infinity,  or  is  parallel  to  U. 

Spatial  Construction. 

14.  In  making  constructions  in  space  we  assume  the 
ability : 

1.  To  draw  through  any  given  point  a  line  parallel  to 
a  given  line. 


14 


SOLID   OR   SPATIAL   GEOMETRY. 


2.  To  pass  a  plane  through  any  given  point  or  line. 

3.  To  make  a  plane  construction,  according  to  the 
principles  of  plane  geometry,  upon  any  assumed  or  deter- 
mined plane. 

Ex.  1.  Problem.  From  a  given  point  without  a  plane 
to  draw  a  normal  to  the  plane. 

Let  P  be  the  point,  and  U 
be  the  plane. 

Con.  Draw  any  line  OB  in 
U,  and  from  P  draw  PO  -L  to 
OB  (P.  Art.  120). 

In   ?7draw   ON  ^_  to  OB; 
and  from  P  draw  PN  1.  to   ON.     PN  is  the  normal 
required. 

For  OB  is,  by  construction,  _L  to  both  OP  and  ON, 
and  therefore  to  the  plane  of  these  lines,  and  hence  to 
PN,  which  lies  in  this  plane  (Art.  9.  Coi-.  2). 

Therefore  PN  is  ±  to  OB  and  to  ON,  and  is  conse- 
quently normal  to  U. 

Ex.  2.  Problem.  To  draw  a  common  perpendicular  to 
two  non-com  planar  lines. 

Let  L,  M  be  the  two  non- 
complanar  lines. 

Con.  In  M  take  any  point, 
A,  and  through  A  draw  the 
line  iV parallel  to  L  (Art.  14. 1). 

M  and  N  determine  a  plane, 
U,    which    is    parallel    to    L. 

From  any  point  B  in  L  draw  BC  normal  to  CT"  (Ex.  1). 
Then,  as  L  is  parallel  to  U,  BC  is  ±  to  L. 


SPATIAL  CONSTRUCTION.  15 

Draw  CD  parallel  to  L  to  meet  M  in  D,  and  from  D 
draw  DE  ±  to  L. 

Then  DE  is  ±  to  both  L  and  ilf,  or  is  their  common 
perpendicular. 

For  DE  is  ±  to  i  by  construction,  and  being  thus 
parallel  to  CB,  EC  is  a  rectangle,  and  ED  is  normal  to 
U,  and  therefore  J_  to  M. 

Cor.  Since  CD  can  meet  M  in  only  one  point,  only 
one  common  perpendicular  can  be  drawn  to  two  nou- 
complanar  lines. 

EXERCISES  A. 

1.  How  many  planes  at  least  determine  one  line  ? 

2.  How  many  lines  at  most  are  determined  by  3  planes  ?  by 
6  planes  ?  by  n  planes  ? 

3.  How  many  planes  at  most  are  determined  by  4  points  ?  by 
8  points  ?  by  n  points  ? 

4.  Draw  a  normal  to  a  plane  from  a  point  in  the  plane. 

5.  Through  one  of  two  non-complanar  lines,  to  pass  a  plane  to 
be  parallel  with  the  other  line. 

6.  Show  that  the  common  perpendicular  to  two  non-complanar 
lines  is  the  shortest  segment  from  one  line  to  the  other. 

7.  From  a  given  point  in  one  of  two  non-complanar  lines,  to 
draw  a  segment  of  given  length  to  meet  the  other.  The  solutions 
are  two,  one,  or  none.    Distinguish  these  cases. 

8.  Given  two  non-complanar  lines,  to  draw  a  segment  from 
one  to  the  other  so  as  to  be  perpendicular  to  one  of  them. 

9.  Given  two  non-complanar  lines,  to  draw  a  segment  from 
one  to  the  other  so  as  to  make  equal  angles  with  each.  Show  that 
this  angle  may  vary  from  a  right  angle  to  the  complement  of  one- 
half  the  angle  between  the  given  lines. 


16  SOLID   OR   SPATIAL   GEOMETRY. 

10.  PO  meets  the  plane  U  (Fig.  of  Art.  12)  at  an  angle  of  30°, 
and  PN  is  normal  to  U.  OA  is  a  planar  line  making  the  angle 
POA  =  60°.     Show  that  cos  AON  =  \  y/Z. 

11.  PO  meets  [/"at  an  angle  a,  and  OiV is  the  projection  of  OP 

on   U.     OA  is  a  planar  line  making  the  angle  POA  =  /3.    Show 

cos/3 

that  cos  A  ON  =  — — — 

cos  a 

12.  Through  the  point,  where  a  given  line  meets  a  plane,  to 
draw  a  planar  line  to  make  a  given  angle  with  the  given  line. 

Examine  the  limits  of  possibility. 


SECTION  2. 

Two  Planes  —  Dihedral  Angle  —  Plane 
Sections. 


15.  Def.  Parallel  planes  are  such  as  meet  only  at 
infinity,  i.e.  which  do  not  meet  at  any  finite  point. 

Cor.  1.  Planes  which  have  a  common  normal  are 
parallel.  For  if  the  planes  meet  at  any  finite  point, 
two  perpendiculars  can  be  drawn  from  that  point  to  the 
same  common  normal,  one  in  each  plane.  But  this  is 
impossible  (P.  Art.  61). 

Cor.  2.  Planes  which  are  not  parallel  intersect  in  a 
line  not  at  infinity.  This  line  is  common  to  the  two 
planes,  and  is  the  common  line  of  the  planes. 

When  two  planes  are  parallel,  their  common  line  is  at 
infinity. 

16.  U  and  V  are  two 

planes  having  AB  as  their 
common  line. 

Prom  any  point,  P,  in 
AB  draw  PC  in  U  and 
PD  in  V,  each  perpendic- 
ular to  AB. 

The  angle  CPD  is  defined  as  the  angle  between  the 
planes  ?7and  V.     Therefore: 

17 


18  SOLID   OR    SPATIAL   GEOMETRY. 

Def.  1.  The  angle  between  two  planes  is  the  angle 
between  two  lines,  one  in  each  plane,  and  both  perpendic- 
ular to  the  common  line  of  the  planes. 

AB  is  normal  to  the  plane  of  PC  and  PD,  and  is 
therefore  perpendicular  to  OF  and  DX  (Art.  9.  Cor.  2). 

Hence,  if  CF  be  ±  to  PD,  and  DX  to  PC,  CF  is 
normal  to  V,  and  DX  is  normal  to  U. 

And  these  normals,  being  complanar,  intersect  in  some 
point,  E,  and  the  angle  CED  is  the  supplement  of  the 
angle  CPD.  Hence,  if  we  consider  GY  and  DX  in  the 
same  sense,  i.e.  from  distal  extremity  to  foot,  or  vice 
versa,  the  angle  CED  is  the  angle  between  the  normals 
to  the  planes,  and  therefore : 

Def.  2.  The  angle  between  two  planes  is  the  supple- 
ment of  the  angle  between  normals  to  the  planes. 

When  CP  is  perpendicular  to  PD,  the  planes  are  per- 
pendicular to  one  another,  and  CP  is  normal  to  V,  and 
DP  to  U.     Hence : 

Def.  3.  Two  planes  are  perpendicular  to  one  another 
when  one  of  them  contains  a  normal  to  the  other. 

17.  Def.  When  PC  is  perpendicular  to  PD,  and 
each  is  perpendicular  to  AB,  the  three  planes  U,  V,  and 
the  plane  of  PCD  are  mutually  perpendicular  to  one 
another.  These  planes  are  then  called  the  rectangular 
co-ordinate  planes  of  space,  and  the  common  point,  P, 
is  the  origin. 

If  we  assume  the  positions  of  these  three  planes,  and 
therefore  the  position  of  the  origin,  the  position  of  any 
point  in  space  can  be  determined  by  giving  its  distances 
from  these  planes,  each  distance  being  affected  with  a 


DIHEDRAL   ANGLE  —  PLANE   SECTIONS.  19 

proper  algebraic  sign.    This  is  the  fundamental  principle 
in  analytic  geometry  of  three  dimensions. 

18.  If  PQ  be  any  line  in  U,  and  PR  be  any  line  in  V, 
meeting  the  common  line  AB,  in  the  same  point,  P,  PQ 
and  PR  are  complanar  (Art.  6.  Cor,  2)  ;  and  if  "W^  denote 
their  plane,  PQ  is  the  common  line  of  U  and  W,  PR 
is  the  common  line  of  W  and  V,  and  AB  is  the  common 
line  of  Fand  U,  and  these  three  lines  are  concurrent 
at  P. 

Therefore,  three  planes,  no  two  of  which  are  parallel, 
and  which  do  not  form  an  axial  pencil,  determine  one 
point,  and  this  point  is  the  point  of  concurrence  of  the 
three  common  lines  of  the  planes  taken  in  twos. 

This  point  is  at  infinity  when  the  three  common  lines 
are  all  parallel. 

Cor.  Three  planes  cannot  form  a  closed  figure.  For 
the  planes  determine,  at  most,  three  concurrent  lines, 
which,  meeting  in  one  common  point,  can  never  meet 
in  any  other  points. 

19.  Def.  When  a  spatial  figure,  S,  is  cut  by  a  plane, 
U,  the  combination  of  elements  common  to  /Sand  f7form 
upon  U  a  plane  figure,  which  is  called  the  plane  section  of 
S  by  U,  or  simply  the  section  of  S  by  U. 

This  definition  suggests  to  us  a  relation  existing  be- 
tween plane  and  spatial  geometry. 

Plane  geometry  may  be  aptly  described  as  a  plane 
section  of  spatial  geometry.  The  plane  upon  which  the 
figures  of  plane  geometry  lie  (P.  Art.  11)  is  the  plane 
of  section,  and  the  figures  themselves  may  be  considered 
as  sections  of  spatial  figures. 


20  SOLID   OR   SPATIAL   GEOMETRY. 

From  this  connection  we  may  be  led  to  expect  that 
relations  existing  among  plane  figures  are  only  particular 
cases  of  more  general  relations  existing  among  spatial 
figures.  And  hence  we  naturally  look  for  many  analogies 
amongst  the  results  of  plane  and  of  spatial  geometry. 

Some  of  these  have  appeared  already,  and  others  will 
present  themselves  in  the  sequel.  And  it  is  worthy  of 
note  how  often  the  number  two  of  plane  geometry  be- 
comes three  in  spatial  geometry.  Thus  two  points  deter- 
mine one  line,  while  three  points  determine  one  plane ; 
two  lines  in  the  plane  determine  one  point,  while  it 
requires  three  planes  to  determine  one  point. 

20.  The  following  theorems  are  self-evident : 

1.  The  section  of  a  line  is  a  point. 

2.  The  section  of  a  plane  is  a  line. 

Hence  spatial  figures  composed  of  lines  and  planes 
give,  in  section,  plane  figures  composed  of  points  and 
lines. 

Def.  Sections  made  by  parallel  planes  are  parallel 
sections. 

21.  Theorem.  Parallel  sections  of  a  plane  are  parallel 
lines. 

Proof.  If  C/'and  U'  be  parallel  planes  which  cut  the 
plane  W,  the  common  lines  ?71Fand  ^J'T^both  lie  in  W, 
and  as  ?7and  U'  meet  only  at  infinity  (Art.  15),  these 
common  lines  meet  only  at  infinity  and  are  parallel. 

Cor.  1.  The  section  of  a  system  of  parallel  planes  is 
a  system  of  parallel  lines. 


PLANE   SECTIONS.  21 

Cor.  2.  The  section  of  an  axial  pencil  is  a  set  of 
parallel  lines  when  the  section-plaue  is  parallel  to  the 
axis ;  in  other  cases  it  is  a  flat  pencil. 

22.  Theorem.  Parallel  sec- 
tions of  two  intersecting  planes 
contain  the  same  angle. 

U  and  V  are  intersecting 
planes,  and  W  and  X  are  two 
parallel  planes  of  section,  the 
sections  being  the  lines  BA, 
BC,  ED,  and  EF. 

Then  AB  is  parallel  to  DE,  (Art.  21.) 

and  BG  is  parallel  to  EF. 

.'.  ZABC=ZDEF. 

Def.  If  W  be  normal  to  the  common  line  of  the 
planes  U  and  V,  the  section  is  called  a  right  section. 
Hence,  the  angle  between  two  planes  is  the  angle  be- 
tween the  two  lines  which  form  the  right  section  of  the 
planes. 

A  system  of  any  number  of  planes  admits  of  a  right 
section  when  all  the  common  lines  of  the  planes  are 
parallel.  In  every  case,  the  term  "  right  section  "  must 
have  reference  to  some  particular  line  or  set  of  parallel 
lines. 


22  solid  oe  spatial  geometry. 

Dihedral  Angle. 

23.  When  we  cut  two  intersecting  planes,  U  and  V, 
by  a  third  plane,  X,  we  get  (Fig.  of  Art.  22), 

1.  A  point  E,  the  vertex  of  the  angle  DEF; 

2.  The  lines  ED  and  EF,  forming  the  arms  of  the 
angle  DEF. 

Now,  1st,  to  the  plane  angle  DEF  corresponds  the 
dihedral  angle  between  the  planes  U  and  F;  and,  2d, 
to  the  vertex  E  of  the  plane  angle  corresponds  the  com- 
mon line,  BE,  of  the  two  planes,  this  line  being  called 
the  edge  of  the  dihedral  angle ;  and,  3d,  to  the  arms  ED 
and  EF  of  the  plane  angle  correspond  the  planes  U  and 
V,  called  the  faces  of  the  dihedral  angle. 

Thus  in  section  a  dihedral  angle  becomes  a  plane  angle, 
the  faces  become  arms,  and  the  edge  becomes  the  vertex. 

If  the  section  be  a  right  section,  the  plane  angle  and 
the  dihedral  angle  have  the  same  measure.  And  as  a 
plane  angle  is  generated  by  rotating  a  line  about  a  point 
in  the  line  taken  as  a  pole  (P.  Art.  32),  so  a  dihedral  angle 
is  generated  by  the  revolution  of  a  plane  about  any  line 
in  the  plane,  taken  as  an  axis. 

The  angular  measurements  are  thus  the  same  for  plane 
and  dihedral  angles. 

24.  Def.  The  plane  which  is  normal  (Art.  9.  Def.  3) 
to  the  join  of  two  given  points  at  its  middle  point,  is  the 
right-bisector  plane  of  the  join  of  the  points. 

Cor.  Since  a  line-segment  has  only  one  middle  point, 
and  a  plane  has  only  one  normal  at  any  given  point,  it 


DIHEDRAL  ANGLE. 


23 


\ 

p 

* 

1  w 

Ai 

C 

k 

u 

N 

follows  that  a  given  line-segment  has  only  one  right- 
bisector  plane. 

A  section  through  the  segment  gives  the  segment  and 
its  right-bisector,  of  plane  geometry. 

25.  Theorem.  Every  point  upon  the  right-bisector 
plane  of  a  segment  is  equally  distant  from  the  end  points 
of  the  segmeut. 

Let  AB  be  a  given  segment, 
and  let  U  be  the  right-bisector 
plane  of  the  segment,  passing 
through  its  middle  point  C,  and 
let  P  be  any  point  on  U.  Then 
P  is  equidistant  from  A  and  B. 

Proof.  Since  A,  B,  and  P  are 
complanar,  let  the  plane  W  pass 
through  these  points.  In  the  section  by  W  "we  have 
the  segment  AB  and  its  right-bisector  CP;  and  hence 
PA^PB  (P.  Art.  53). 

It  will  be  here  noticed  that  the  proof  is  obtained 
immediately  by  reducing  the  theorem  to  depend  upon 
the  corresponding  one  in  plane  geometry. 

In  like  manner  we  readily  prove  the  converse : 

Every  point  equidistant  from  the  end  points  of  a  given 
line-segment  is  upon  the  right-bisector  plane  of  the  seg- 
ment. 

Cor.  From  this  it  appears  that  the  locus  of  a  point 
which  is  equidistant  from  two  fixed  points  is  the  right- 
bisector  plane  of  the  join  of  the  points. 

26.  Def.  The  planes  which  pass  through  the  edge  of 
a  dihedral  angle  and  make  equal  angles  with  its  faces 
are  the  bisectors  of  the  dihedral  angle. 


24  SOLID   OR   SPATIAL  GEOMETRY. 

The  proofs  of  the  following  theorems  may  be  obtained 
at  once  by  making  them  to  depend  upon  the  correspond- 
ing theorems  in  plane  geometry. 

1.  The  two  bisectors  of  a  dihedral  angle  are  perpen- 
dicular to  one  another. 

For  proof,  make  a  right  section  of  the  dihedral  angle 
and  apply  (F.  Art.  45). 

2.  Any  point  upon  a  bisector  of  a  dihedral  angle  is 
equally  distant  from  the  faces  of  the  angle. 

For  proof,  make  a  right  section  through  the  point  and 
apply  (P.  Art.  68). 

3.  Any  point  equidistant  from  the  faces  of  a  dihedral 
angle  is  on  one  of  the  bisectors  of  the  angle. 

Proof  as  in  2. 

27.  Theorem.     Any  two  lines   are  divided  similarly 

(P.  Art.  201.  Def.)  by  a  system 

of  parallel  planes. 

L  and  M  are  two  lines  cut  . 

/      \ —  f^ 

by  the  parallel  planes  U,  V,  and       /jj 
W.    Then  L  and  iltf  are  similarly 


divided. 

Proof.     A,  B,  C  and  A',  B',      /^     ^  ^ "f 
C"  are  corresponding  points  of  / 

sectionofthe  two  lines.  Through       /    c^^-^^^q     _,\^, 

A  draw  the  line  N  parallel  to      '^   i   — -r p 

L,  and  let  it  meet  the  planes  at  /         J  '■- 

A,  P,  and  Q.  ' 

Then  L  and  iV  being  complanar  (Art.'  6.  Cor.  4), 
AA'  is  II  to  PB'  is  II  to  QC"; 
.-.  A'B'  =  AP,  and  B'O  =  PQ. 


/ 


DIHEDRAL   ANGLE. 


25 


But  ACQ  is  a  triangle,  and  BP  is  parallel  to  CQ', 

.-.  AB:BC=AP:PQ  =  A'B':B'C'. 

Or  the  lines  L  and  M  are  similarly  divided. 

Cor.  1.  The  parallel  planes  of  a  system  divide  all 
lines  similarly. 

Co7'.  2.  The  segments  of  parallel  lines  intercepted 
between  the  same  two  parallel  planes  are  equal. 

28.  Theorem.  If  three  concurrent  non-complanar  lines 
be  divided  similarly  in  relation  to  the  point  of  concur- 
rence, the  triplets  of  corresponding  points  determine  a 
system  of  parallel  planes. 

L,  M,  N  are  three  non-com- 
planar  lines  concurrent  at  O, 
and  are  divided  at  A,  B,  C,  •••, 
A',  B',  C, ...,  and  A",  B",  C"  ■-, 
so  that 

OA:AB:BC=  OA' :  A'B' :  B'C 
=  OA"  :  A"B"  :  B"C". 

Then   the    planes   determined  l 
by  AAA",  BB'B",  CC'C",  etc., 
are  parallel. 

Proof.     AA'  is  II  to  BB'  is  II  to  CC, 

and  ^ A "  is  II  to  BB"  is  II  to  CC".    (P.  Art.  202.  Cor. ) 

Let  OP  be  normal  to  the  plane  AAA".  Then  OP  is 
±  to  A  A'  and  AA"  (Art.  9.  Cor.  2),  and  therefore  to  BB' 
and  BB",  and  to  CC  and  CC". 


26  SOLID   OR,   SPATIAL  GEOMETRY. 

Hence  OP  is  normal  to  the  planes  BB'B"  and  CC'G", 
and  the  three  planes  AA'A",  BB'B",  CC'C"  are  accord- 
ingly parallel  (Art.  15.  Cor.  1). 

Cor.  1.  Since  AA'  is  II  to  BB',  and  AA"  is  11  to  BB", 
etc.,  the  A  AA'A",  BB'B",  and  CC'C"  are  similar.  But 
the  concurrent  lines  L,  M,  N  determine  three  planes 
whose  common  point  is  0;  therefore  parallel  sections 
of  three  non-parallel  planes  are  similar  triangles. 

Cor.  2.  Since  any  polygon  may  be  divided  into  triangles, 
and  similar  polygons  into  similar  triangles  similarly 
placed  (P.  Art.  206),  it  follows  that : 

Parallel  sections  of  any  number  of  planes  having  a 
common  point  are  similar  polygons. 

29.  Def.  Four  non-complanar  lines  which  intersect 
two  and  two  in  four  points,  form  a  skew-,  or  a  gauche-,  or  a 
spatial  quadrilateral. 

The  sides  of  the  skew  quadrilateral  and  its  two  diag- 
onals are  six  lines  connecting  four  points  in  space,  and 
form  the  six  edges  of  a  figure,  to  be  described  hereafter, 
called  the  Tetrahedron. 

The  skew  quadrilateral  is  a  plane  quadrilateral  with 
one  vertex,  and  the  sides  forming  it  raised  out  of  the 
plane. 

30.  Theorem.  The  joins  of  the  middle  points  of  the 
opposite  sides  of  a  skew  quadrilateral  bisect  one  another. 

ABCD  is  a  skew  quadrilateral,  AB  and  BC  lying  in 
a  plane  different  from  the  plane  of  CD  and  DA.  AC 
and  BD  are  the  diagonals. 


SKEW   QUADRILATERAL. 


27 


E,  F,  G,  H  are  middle  points  of  the  sides  upon  which 
they  lie.     Then  EG  and  FH  bisect  one  another. 

Proof.  EF  and  GH  are 
both  parallel  to  AC,  and 
equal  to  half  AC  (P.  Art. 
202);  they  are  therefore  equal 
and  parallel  to  one  another. 

Therefore  EFGII  is  a  par- 
allelogram, and  its  diagonals 
EG  and  FH  bisect  ,one  an- 
other (P.  Art.  81.  3). 

Cor.  1.  Let  /  and  J  be  the  middle  points  of  the 
diagonals  AC  and  BD. 

Then  ACBD  is  a  skew  quadrilateral,  and  the  joins  of 
middle  points  of  opposite  sides  are  FH  and  IJ. 

Therefore  FH  and  IJ  bisect  one  another ;  and  hence 
FH,  IJ,  and  EG  mutually  bisect  each  other. 

Cor.  2.  A,  B,  C,  D  are  four  points  in  space,  and  AB, 
AC,  AD,  BC,  BD,  and  CD  are  their  six  connectors. 

Therefore  if  four  points  in  space  be  connected  two  and 
two  by  six  line-segments,  the  joins  of  the  middle  points 
of  these  connectors  taken  in  opposite  pairs  are  concurrent, 
and  mutually  bisect  one  another. 


EXERCISES  B. 

1.  Draw  a  line  equally  inclined  to  two  intersecting  planes.     Is 
the  problem  definite  or  indefinite  ? 

2.  If   U  and  V  be  two  planes,  and   U  contains  a  normal  to  V, 
show  that  V  contains  a  normal  to  U. 


28  SOLID   OR   SPATIAL   GEOMETRY. 

3.  Two  lines  may  be  drawn,  one  on  each  of  two  intersecting 
planes,  so  as  to  make  an  angle  with  one  another  of  any  magnitude 
from  zero  to  a  straight  angle. 

4.  If  three  concurrent  non-complanar  parallel  lines  be  divided 
homographically,  the  planes  determined  by  the  triplets  of  corre- 
sponding points,  all  pass  through  a  common  line.  When  is  this 
line  at  infinity  ? 

6.  If  the  sides  of  a  skew  quadrilateral  are  equal,  the  diagonals 
are  perpendicular  to  one  another. 

6.  What  theorem  is  obtained  from  30  by  bringing  D  to  the 
plane  ot  ABC7 

7.  Draw  the  shortest  path  from  one  point  to  another  so  as  to 
touch  a  given  plane  in  its  course,  both  points  being  upon  the  same 
side  of  the  plane. 

8.  Show  that  a  skew  quadrilateral  cannot  have  four  right  angles. 
How  many  can  it  have  ? 

9.  A,  B,  C,  D  are  four  non-complanar  points.  Show  that  the 
locus  of  a  point  which  is  equidistant  from  A  and  B^  and  also  equi- 
distant from  C  and  Z>,  is  a  line  perpendicular  to  both  AB  and  CD. 

10.  If  A,  B,  C,  D,  E,  F  be  any  6  points  in  space,  a  point  can  be 
found  which  is  equidistant  from  A  and  B,  equidistant  from  C  and 
D,  and  equidistant  from  E  and  F. 


SECTION   3. 

Sheaf  of  Lines  and  Planes  —  Solid  Angle  or 
Corner. 

31.  Def.  Three  or  more  non-complanar  lines  meeting 
in  a  point  form  a  sheaf  of  lines,  and  three  or  more  planes 
passing  through  a  common  point  form  a  sheaf  of  planes. 
The  common  point  is  in  each  case  called  the  centre  of 
the  sheaf. 

The  lines  and  planes  which  form  a  sheaf  pass  through 
the  centre  and  extend  indefinitely  outwards  from  it,  but 
usually  we  have  to  consider  only  those  portions  which 
lie  upon  one  side  of  the  centre,  and  the  centre  is  then 
commonly  called  the  vertex  or  apex  of  the  figure. 

In  a  sheaf  of  lines  the  determined  planes  form  a  sheaf 
of  planes,  and  in  a  sheaf  of  planes  the  determined  lines 
f  rm  a  sheaf  of  lines.  So  that  practically  a  sheaf  of 
lines  and  a  sheaf  of  planes  are  only  the  same  figure 
differently  viewed. 

Cor.  From  Article  27  it  follows  that  the  lines  of  a 
sheaf  are  similarly  divided  by  a  system  of  parallel  planes. 

And  from  Article  28  it  follows  that  if  a  sheaf  of  three 
lines  has  its  lines  similarly  divided  with  reference  to 
the  centre,  the  triplets  of  corresponding  points  deter- 
mine a  set  of  parallel  planes. 

32.  A  non-central  section  of  a  sheaf  of  lines  and  the 
determined  planes  is  a  set  of  points  with  their  determined 

29 


80  SOLID   OR   SPATIAL   GEOMETRY. 

lines ;  and  the  non-central  section  of  a  sheaf  of  planes 
and  the  determined  lines  is  a  set  of  lines  with  their 
determined  points. 

Thus  the  reciprocity  between  a  sheaf  of  lines  and  a 
sheaf  of  planes  is  analogous  to  that  between  a  set  of 
points  and  a  set  of  lines  in  plane  geometry. 

33.  Def.  If  the  points  in  the  section  of  a  sheaf  of 
lines  be  so  disposed  as  to  form  the  vertices  of  a  polygon 
without  re-entrant  angles,  and  only  those  planes  of  the 
sheaf  be  considered,  which,  in  the  section,  form  the 
sides  of  the  polygon,  the  combination  of  lines  and 
planes  in  the  sheaf  forms  a  solid  angle,  or  a  polyhedral 
angle,  or  a  corner. 

L,  M,  N,  Kis  a,  sheaf  of  four  lines  with  centre  0.  Let 
the  sheaf  be  cut  by  the  plane  U,  giving  in  section  the 
points  A,  B,  C,  D  correspond- 
ing to  L,  M,  N,  K,  respectively. 
If  the  polygon  ABCD  is  with- 
out re-entrant  angles,  the  figure 
formed  by  the  lines  L,  M,  N, 
K,  and  the  portions  of  deter- 
mined planes,  LOM,  MON, 
NOK,  KOL,  intercepted  be- 
tween these  lines,  is  a  solid 
angle,  or  a  corner. 

0  is  the  vertex  of  the  corner,  L,  M,  N,  K,  forming 
the  edges  or  axes  of  the  dihedral  angles  are  its  edges ; 
the  planes  LOM,  MON,  NOK,  and  KOL  are  its  faces; 
and  the  angles  LOM,  MON,  NOK,  and  KOL  are  its 
face-angles. 

The  term  corner  or  solid  angle  does  not  involve  any 


SOLID   ANGLES.  31 

particular  length  of.  line,  or  extent  of  plane,  or  magni- 
tude of  angle.  It  involves  the  existence  of  a  number 
of  lines  forming  edges,  with  the  same  number  of  planes 
limited  by  these  lines  and  forming  faces,  and  all  meeting 
at  a  common  point  to  form  a  vertex. 

34.  A  corner  may  have  any  number  of  faces  greater 
than  three,  and  the  same  number  of  edges.  The  one 
figured  in  the  preceding  article  is  a  four-faced  corner, 
or  a  tetrahedral  angle. 

A  section  of  a  three-faced  corner  is  a  triangle;  and 
as  the  triangle  is  the  most  important  of  all  polygons,  so 
the  three-faced  corner,  or  trihedral  angle,  is  the  most 
important  of  all  corners. 

A  corner  will  be  indicated  by  writing  its  vertex 
followed  by  a  point,  and  then  the  letters  indicating 
points  upon  its  several  edges. 

Thiis  the  symbol  0  •  ABGD  denotes  the  four-faced 
corner  as  figured  in  the  preceding  article. 


32 


SOLID   OR   SPATIAL  GEOMETRY. 


Properties  of  Trihedral  Angles,  or   Three- 
faced  Corners. 

35.   Theorem.     In  any  three-faced  corner  the  sum  of 
any  two  face  angles  is  greater  than  the  third. 

0  •  ABO  is  the  three-faced  corner. 

Proof.  If  the  face  angles  are  all 
equal  to  one  another,  the  truth  of  the 
theorem  is  evident.  If  they  are  une- 
qual, let  the  angle  LOJ^  be  >  than 
LOM. 

In  the  plane  of  L  and  N  draw  OK, 
making  the  angle  LOK=  LOM,  and  on 
M  and  K  take  OB  =  OD  =  any  conven- 
ient length,  and  let  A  be  any  point  on  L,  other  than  O. 
Let  the  plane  of  ABD  cut  N  in  C. 

Then  AAOB  =  AAOD.  (P.  Art.  52.) 

.-.  AD  =  AB,  and  Z  ADB  =  Z  ABD. 

.:  Z  CDB  is  >  Z  CBD,  and  CB  is  >  CD.    (P.  Art.  62. 2.) 

But  in  the  A  BOO  and  DOC,  BO  =  DO  by  construc- 
tion, OC  is  common,  and  BC  >  CD. 

.:  Z  BOO  is  >  Z  DOC,  (P.  Art.  67.) 

and      •.•  Z  AOB  —  Z  AOD  by  construction, 

.-.  ZAOB  +  A  BOC  is  >  Z  AOD  +  Z  DOC. 
Or  Z  ^OJS  -f-  Z  BOC  is  >  Z  ^OC. 


TRIHEDRAL   ANGLE   OR  THREE-FACED  CORNER.      33 

36.  Problem.  To  find  the  locus  of  a  point  equidistant 
from  the  three  edges  of  a  three-faced  corner. 

O  is  the  vertex,  and  L,  M,  N  the  edges  of  the  three- 
faced  corner.  ^ 

Let  P  be  a  point  on  the  required 
locus,  and  PA,  PB,  PC  be  perpendic-  > 

ulars  upon  the  edges  L,  M,  and  N  / 

respectively.  /  / 

In  the  right-angled  triangles  POA,     /y^ 
POB,    POC,    PO  is   a  common  hy-   /     L 
pothenuse,    and    PA  =  PB  =  PC  by         / 
hypothesis. 

Therefore  the  triangles  are  congruent,  and  OA  =  OB 
=:  OC.  And  the  circle  through  A,  B,  C  is  Si  cone  circle 
with  0  and  P  as  two  vertices. 

Therefore  OP  passes  through  the  centre  of  this  circle 
and  is  normal  to  its  plane. 

Hence  the  construction :  take  OA  =  OB  =  OC  and  join 
0  with  the  centre  of  the  circle  through  A,  B,  and  C; 
this  join  is  the  locus  required. 

Def.  The  locus  just  found  is  a  line  equally  inclined 
to  the  three  edges,  and  is  an  isoclinal  line  to  the  edges. 
A  plane  normal  to  this  line  is  also  equally  inclined  to 
the  edges  and  is  an  isoclinal  plane  to  the  edges. 

Cor.  Since  the  edges  may  be  considered  as  indefinite 
lines  extending  through  the  vertex  and  forming  a  sheaf 
of  three,  the  three  measures  OA,  OB,  OC  may  each  be 
taken  in  two  opposite  directions,  or  we  can  have  eight 
variations  of  sign  in  all.  But  four  of  these  are  the  other 
four  reversed. 

Therefore  three  lines  forming  a  sheaf  have  four  iso- 
clinal lines  and  four  isoclinal  planes  through  the  centre. 


34 


SOLID   OR   SPATIAL   GEOMETRY. 


37.  Def.  Corners  are  equal  when  they  can  be  super- 
imposed so  as  to  form  virtually  but  one  corner.  In  this 
superposition  the  vertices  coincide,  and  the  edges  coincide 
in  pairs,  one  from  each  corner. 

38.  Theorem.  Two  three-faced  corners  are  equal  when 
the  face  angles  of  the  one  are  respectively  equal  to  the 
face  angles  of  the  other,  and  they  are  disposed  in  the 
same  order  about  the  vertices.  0  •  LMN  and  0'  •  L'M'N' 
are  two  three-faced  corners  having  Z.  LOM=  Z.  L'O'M', 
ZMON=zZM'0'N',  ZNOL  =  ZN'0'L',  and  having 
these  disposed  in  the  same  order  about  the  vertices ;  i.e. 
so  that  the  order  of  magnitude  of  the  angles  is  according 
to  the  same  species  of  rotation  for  each.  Then  the 
corners  are  equal. 

Proof.  Take  OA=OB=OC=  O'A'  =  O'B'  =  O'C,  A 
and  A'  being  on  corresponding  edges,  etc. 

The        AAOB  =  AA' O'B',  and  ^B  =  A'B'. 
Similarly,        BC  =  B'C,  and  CA  =  C'A', 
and  the  A  ABC  =  A  A' B'C. 

Therefore  when  A' B'C  is  superimposed  on  ABC,  the 
centres  of  their  circumcircles 
coincide,  and  the  normals  to 
the  planes  of  these  circles  at 
their  centres  coincide,  and 
hence  the  vertices  of  the 
corners,  lying  on  these  nor- 
mals, coincide  (Art.  36),  and 
the  two  corners,  coinciding  in 
all  their  parts,  form  virtually 
but  one  corner. 


TRIHEDRAL  ANGLE  OR   THREE-FACED  CORNER.      35 

39.  Two  triangles  may  be  congruent  and  yet  not  be 
superposable  until  one  of  them  is  turned  over  in  the  plane. 
This  operation,  which  is  possible  and  allowable  in  plane 
geometry,  is  not  always  practicable  in  spatial  geometry. 

Suppose  the  two  three-faced  corners  of  the  previous 
article  to  be  so  placed  that  the  triangles  ABC  and 
A'B'C  lie  in  one  plane,  and  0  and  0'  are  upon  the  same 
side  of  this  plane.  Then  the  triangles  are  directly 
superposable  and  the  corners  are  superposable  and  equal. 

But  if  the  triangles  ABC  and  A'B'C  be  in  the  same 
plane  and  be  directly  superposable  while  0  and  0'  are 
upon  opposite  sides  of  the  plane,  or  if  0  and  0'  be  upon 
the  same  side  of  the  plane  while  the  triangles  are  not 
superposable  until  one  of  them  is  turned  over  in  the 
plane,  then  the  two  corners,  although  having  correspond- 
ing parts  respectively  equal,  are  not  superposable,  and 
are  not,  therefore,  equal  according  to  definition. 

A  little  consideration  will  show  that  in  the  non-super- 
posable  case,  the  face  angles  are  disposed  in  opposite 
orders  about  the  vertices  of  the  two  corners. 

Def.  Two  three-faced  corners  having  corresponding 
parts  respectively  equal  but  not  being  superposable  are 
said  to  be  equal  by  symmetry,  or  to  be  symmetrical^  to 
each  other. 

Symmetrical  figures  are  related  to  each  other  in  the 
same  manner  as  an  object  and  its  image  in  a  plane 
mirror,  or  as   the   right  and   the   left  hand;  and  they 

1  The  term  conjugate  and  opposable  have  both  been  employed  to 
express  the  condition  here  described.  But  it  is  obvious  that  the  well- 
known  term  symmetrical  expresses  exactly  what  is  meant,  and  can- 
not therefore  be  profitably  superseded  by  any  other  word. 


36  SOLID    OK   SPATIAL   GEOMETRY. 

might  be  called  right-handed  and  left-handed  figures  if 
there  were  any  means  of  distinguishing  between  which 
should  be  called  right-handed,  and  which  left-handed. 
In  certain  parts  of  crystallography  the  means  of  distin- 
guishing is  apparent,  and  this  terminology  is  employed. 

Two  superposable  figures  can  be  in  perspective  with 
respect  to  a  centre  at  infinity,  while  two  symmetrical  fig- 
ures can  be  in  perspective  with  respect  to  a  centre  which 
is  the  middle  point  of  the  joins  of  corresponding  parts. 

Cor.  It  is  readily  seen  that  two  n-faced  corners  may 
be  superposable  and  equal,  and  also  that  they  may  be 
symmetrical  and  not  superposable. 

But  where  there  are  more  than  three  faces,  new  possi- 
bilities arise,  for  the  face  angles  may  be  equal  in  number 
and  respectively  equal  in  magnitude,  and  yet  the  corners 
may  be  neither  equal  nor  symmetrical. 

40.  Theorem.  Of  two  dihedral  angles  of  a  three- 
faced  corner  and  the  opposite  face  angles, 

1.  The  greater  face  angle  is  opposite  the  greater 
dihedral  angle ; 

2.  The  greater  dihedral  angle  is  opposite  the  greater 
face  angle. 

0  •  LMN  is  a  three-faced  cor- 
ner having  0  as  vertex,  and  L, 
M,  N  as  edges. 

From  A,  any  point  in  L, 
draw  AB  l.to  M  and  AC  ±  to 
N,  and  from  B  and  C  draw,  in 
the  plane  MN,  perpendiculars 
to  M  and  N  respectively,  and  let  these  perpendiculars 
meet  in  D.     Join  OD. 


TRIHEDRAL   ANGLE   OR   THREE-FACED  CORNER.      37 

The  angles  ABD  and  ACD  are  respectively  the  meas- 
ures of  the  dihedral  angles  whose  edges  are  M  and  N 
(Art.  16.  Def.  1).  The  A  ADB  and  ADG  are  right-angled 
at  D  and  have  AD  as  a  common  side,  the  triangles  ABO 
and  ACO  are  right-angled  at  B  and  C,  and  have  ^0  as  a 
common  hypothenuse,  and  the  A  DOB  and  DOO  are 
right-angled  at  B  and  C,  and  have  OD  as  a  common 
hypothenuse. 

1.  ljetZABDhe>ZACD;  t}ienZAOCis>ZAOB. 
Proof.     Since  Z  ABD  is  >  Z  ACD, 

therefore,  Z  BAD  is  <  Z  C^D,  and  BD  is  < CD; 

and  .-.  BO  is  >  CO,  and  AC  is  >  ^£, 

and  .-.  ZAOC  is  >ZA  OB. 

2.  This,  which  is  the  converse  of  1,  follows  from  the 
law  of  Identity  (P.  Art.  7). 

Cor  1.  If  a  three-faced  corner  has  two  dihedral  angles 
equal,  it  has  two  face  angles  equal ;  and  conversely,  if 
it  has  two  face  angles  equal,  it  has  two  dihedral  angles 
equal. 

Cor.  2.  A  three-faced  corner  with  three  equal  dihe- 
dral angles  has  three  equal  face  angles,  and  conversely. 

Cor.  3.  If  A,  B,  G  denote  the  dihedral  angles,  and 
a,  b,  c  denote  the  opposite  face  angles,  the  order  of  mag- 
nitude is  the  same  for  A,  B,  C,  and  a,  b,  c. 

It  will  be  noticed  that  in  this  theorem  and  its  corol- 
laries the  relations  between  the  dihedral  angles  and  face 
angles  are  analogous  to  those  between  the  angles  and 
sides  of  a  plane  triangle. 


88 


SOLID   OR   SPATIAL   GEOMETRY. 


Def.  A  three-faced  corner  with  its  edges  mutually 
perpendicular  to  one  another  is  a  rectangular  corner  or 
a  right  corner.  It  has  all  its  dihedral  angles  right  angles, 
and  all  its  face  angles  right  angles. 

41.  Problem.  Being  given  the  face  angles  of  a  three- 
faced  corner,  to  construct  plane  angles  which  shall  have 
the  same  measures  as  the  dihedral  angles. 

0  •  LMN  is  the  given  three-faced  corner.  To  draw  a 
plane  angle  which  shall  have  the  same  measure  as  the 
dihedral  angle  whose  edge  is  L. 

Constr.  Through  A,  any  point  in  L,  draw  a  plane 
normal  to  L,  and  cutting  M and  Nin  B  and  C. 


(1)  (2) 

In  (2)  take  O'A'  =  OA,  and  through  A'  draw  a  line, 
K,  perpendicular  to  O'A'.  Draw  O'B',  making  the 
Z  A'0'B'  =  ZAOB,  and  O'C,  making  the  angle  A'O'C 
=  ZAOC.  Also,  draw  O'B"  =  O'B'  and  making  the 
ZC'0'B"=COB.     Join  C'B". 

The  A  DA'C  constructed  with  B'A',  A'C,  and  C'B" 
as  sides,  has  the  angle  C'A'D  equal  in  measure  to  the 
dihedral  angle  whose  edge  is  L. 


TRIHEDEAL   ANGLE   OR   THREE-FACED  CORNER.      39 

Proof.  Since  L  is  normal  to  the  plane  of  AB  and  AC, 
the  Z  BAG  measures  the  dihedral  angle  at  L  (Art.  16. 
Def.  1).  And  in  the  construction  O'B'  =  OB  and  O'C  = 
00,  and  hence  A'B'  =  AB  and  A'C  =  AC;  and  also  we 
have  made  0'B"C'  congruent  with  OBC. 

Hence  the  A  DA'C  is  congruent  with  BAC,  and  the 
Z  DA'C  measures  the  dihedral  angle  at  L. 

Similarly,  the  other  dihedral  angle  may  be  found. 

Cor.  1.  Since  in  the  foregoing  construction  only  one 
triangle  is  possible  with  the  given  elements,  the  dihedral 
angles  of  a  three-faced  corner  are  completely  given  when 
the  face  angles  are  given;  and  hence  the  measures  of  the 
dihedral  angles  are  expressible  in  terms  of  those  of 
the  face  angles. 

Cor.  2.  A  three-faced  corner  is  given  when  its  face 
angles  and  their  order  with  respect  to  the  vertex  are 
given. 

In  w-faced  corners  where  n  is  greater  than  3,  the  giving 
of  the  face  angles  does  not  determine  the  dihedral  angles, 
and  does  not  therefore  determine  the  form  of  the  corner. 
We  have  the  analogue  of  this"  in  plane  geometry,  where 
the  giving  of  the  sides  of  a  polygon,  with  more  than 
three  sides,  does  not  determine  the  form  of  the  polygon. 

In  general,  corners  of  more  than  three  faces  are  not 
of  much  importance  unless  they  are  regular, 

Def.  A  regular  corner  has  all  its  face  angles  equal 
and  all  its  dihedral  angles  equal. 

42.  Theorem.  In  any  corner  the  sum  of  the  face 
angles  is  less  than  a  circumangle. 


40 


SOLID   OR   SPATIAL   GEOMETKY. 


Proof.  Let  the  corner  have  n  faces.  Cut  it  by  a  plane, 
and  we  have,  as  section,  a  polygon  of  n  sides,  the  sum 
of  whose  internal  angles  is  2  (n— 2)  ~\s. 

Denote,  in  general,  a  basal  angle  of  one  of  the  result- 
ing triangular  faces  by  B,  and  a  face  angle  by  F. 

At  each  vertex  of  the  polygonal  section,  three  faces 
meet  to  form  a  three-faced  corner,  viz.  the  section  itself 
and  two  faces  of  the  original  corner. 

.-.  2-B  is  >  the  sum  of  the  internal  angles  of  the 
section,  i.e.  >2{n-  2)  Is.  (Art.  35.) 

But  25  4-2F=2?i1s. 

.-.  Si^is  <4  1s. 

Or  the  sum  of  the  face  angles  is  less  than  a  circumangle. 

43.  Let  0  •  ABC  be  a  three-faced  corner,  and  let  PS 
be  normal  to  the  plane  AOB,  PR  normal  to  the  plane 
COA,  and  PQ  normal  to  the 
plane  BOC. 

The  angle  QPR  is  the  sup- 
plement of  the  dihedral  angle 
at  OC,  EPS  is  the  supple- 
ment of  the  dihedral  angle  at 
OA,  and  SPQ  is  the  supple- 
ment of  the  dihedral  angle  at 
OB  (Art.  16.  Def.  2). 

Therefore  P-QRS  is  a  three-faced  corner  in  which 
the  face  angles  are  supplementary  to  the  dihedral  angles 
of  0  •  ABC. 

Also,  since  OB  is  normal  to  the  plane  SPQ,  etc.,  the 
face  angles  of  0  •  ABC  are  supplementary  to  the  dihe- 
dral angles  of  P-  QRS. 


RECIPROCAL  CORNERS.  41 

Similar  reasoning  will  apply  to  a  corner  of  any  number 
of  faces. 

Def.  Corners  so  related  that  the  dihedral  angles  in 
the  one  are  supplementary  to  the  face  angles  in  the  other 
are  called  reciprocal  corners. 

Cor.  1.  Employing  the  notation  of  Art.  40.  Cor.  3, 
for  one  of  the  corners,  and  the  letters  accented  for  the 
other,  we  have 

^  +  a'  =  -B  +  6'=  C  +  c'=  ... 

=  ^'  +  a=jB'+6  =  C"+ c=...=2ns. 
Now,  in  any  corner   a'4-6'4-c'H — is<41s;   (Art.  42) 

and    A-irB+C-\ \-  a'  +  h'  +  d  +  -••  =  2n~\s, 

where  n  denotes  the  number  of  faces. 
.-.  A-\-B+C+—  is  >(2n-4)ns. 

That  is,  the  sum  of  the  dihedral  angles  of  any  corner 
is  greater  than  the  difference  between  twice  as  many 
right  angles  as  the  figure  has  faces,  and  a  circumangle. 

Cor.  2.  Making  n  =  3,  we  see  that  the  sum  of  the 
dihedral  angles  of  a  three-faced  corner  is  greater  than 
two  right  angles  and  less  than  six  right  angles. 

44.  Problem.  Given  the  dihedral  angles  of  a  three- 
faced  corner,  to  construct  the  face  angles. 

Constr.  Take  the  supplements  of  the  given  dihedral 
angles,  and  considering  these  as  face  angles,  construct 
the  corresponding  dihedral  angles  by  Art.  41.  The  sup- 
plements of  these  latter  angles  are  the  required  face 
angles. 

This  construction  is  evident  from  the  preceding  article. 


42  SOLID   OR    SPATIAL   GEOMETRY. 

Cor.  1.  It  is  readily  seen  that  only  one  set  of  face 
angles  can  be  obtained  when  a  set  of  dihedral  angles  is 
given ;  so  that  when  the  dihedral  angles  of  a  three-faced 
corner  are  given,  the  face  angles  are  given  also  ;  and  the 
face  angles  can  be  expressed  in  terms  of  the  dihedral 
angles. 

Cor.  2.  A  three-faced  corner  is  given  when  the  dihe- 
dral angles,  and  their  order,  are  given. 

To  construct  a  three-faced  corner  when  its  face  angles 
are  given  is  analogous  to  constructing  a  triangle  when 
its  sides  are  given ;  and  to  construct  the  corner  when  its 
dihedral  angles  are  given  is  analogous  to  constructing 
the  triangle  when  its  angles  are  given.  And  this  latter 
is  a  definite  problem  with  respect  to  the  corner,  but  an 
indefinite  one  with  respect  to  the  triangle. 

45.  Problem.  To  find  the  locus  of  a  point  equidistant 
from  three  given  points  not  in  line. 

Let  A,  B,  C  be  the  points,  and  let  U  be  the  right- 
bisector  plane  of  AB,  and  V  be  the  right-bisector  plane 
of  ^a  (Art.  24  Def.). 

Every  point  equidistant  from  A  and  B  is  on  U  (Art. 
25.  conv.),  and  every  point  equidistant  from  A  and  C  is 
on  V.  And  the  required  locus  is  the  common  line  of  U 
and  V.  But  this  line  evidently  passes  through  the  cir- 
cumcentre  of  the  triangle  ABC  and  is  normal  to  its  plane. 

Hence  the  locus  of  a  point  equidistant  from  three  given 
points,  not  in  line,  is  the  axis  of  vertices  of  the  circum- 
circle  of  the  three  points  considered  as  a  cone-circle. 

Cor.  The  three  right-bisector  planes,  of  the  joins  of 
three  points,  taken  two  and  two,  form  an  axial  pencil. 


c 


LOCI.  43 

46.  Problem.  To  find  a  point  equidistant  from  four 
given  points  which  are  not  complanar,  and  no  three  of 
which  are  in  line. 

Let  A,  B,  C,  D  be  the  four  points,  and  let  PO  be  the 
locus  of  a  point  equidistant  from 
A,  B,  and  C.     Join  D,  the  fourth 
point,  to  any  one  of  the  other 

three,  as  C,  and  draw  the  right-    

bisector  plane,  X,  of  CD.  ^\^ 

As  D  is  not  complanar  with  ^^v^     "^      /^^^ 

A,  B,  and  C,  the  plane  X  is  not 
parallel   to  PO,   and    therefore  \ 
meets  PO  at  some  point  0.     But        , 
0  is  equidistant  from  A,  B,  and 
C,  and  it  is  also  equidistant  from  C  and  D. 

Therefore  0  is  equidistant  from  A,  B,  C,  and  D. 

Cor.  1.  The  line  OP  is  the  common  line  to  three 
bisector  planes,  namely,  those  of  AB,  BC,  and  CA  (Art. 
45.  Cor.),  and  X  is  a  fourth  plane  which  goes  through 
the  point  0.  The  two  remaining  bisector  planes,  those 
of  AD  and  CD,  must  pass  through  the  same  point  0. 

Therefore  the  six  right-bisector  planes  of  the  joins  of 
four  non-complanar  points,  of  which  no  three  are  in  line, 
pass  through  a  common  point  and  form  a  sheaf  of  planes. 

Cor.  2.  The  four  points  can  be  combined  to  form  four 
different  triangles,  and  the  lines,  such  as  PO,  which  pass 
through  their  circumcentres  and  are  normal  to  their 
planes,  all  pass  through  0  and  form  a  sheaf  of  lines. 

Cor.  3.  As  the  line  PO  can  meet  the  plane  X  in  only 
one  point,  there  can  be  only  one  point  equidistant  from 
A,  B,  C,  and  D. 


44  SOLID   OR   SPATIAL   GEOMETRY. 


EXERCISES  C. 

1.  Any  face  angle  of  a  three-faced  corner  is  greater  than  the 
difference  between  the  other  two. 

2.  Show  how  to  construct  a  corner  symmetrical  with  a  given 
corner. 

3.  Show  that  the  three  bisectors  of  the  dihedral  angles  of  a 
three-faced  comer  have  a  common  line,  and  that  this  line  is  an 
isoclinal  to  the  three  faces. 

4.  There  are  four  isoclinal  lines  through  the  vertex  to  the  three 
faces  of  a  three-faced  corner. 

6.    In  the  figure  (2)  of  Art.  41  denote  O'A'  by  p. 
Then    A'D  =  A'B'  =pidinc;  O'B"  =  O'B'  -p sec c; 

A'C  =ptan6;  0' C  =psecb; 

and 

C'D^  =  C'B"-^  =  DA'^  +  A'C"^  -  2  DA'  •  A'C  cos  A    (P.  Art.  217.) 

=  O'C"^  +  O'B"^  -2a  C-  aB"  cos  a. 

.:  substituting,  and  dividiag  by  j)'^, 

tan2  c  -t-  tan2  6  —  2  tan  c  •  tan  b  cos  A 

=  sec^  c  +  sec2  6  —  2  sec  c  •  sec  b  cos  a, 

whence  by  reduction  and  dividing  by  cos  b  cos  c, 

cos  a  =  cos  b  cos  c  -|-  sin  6  sin  c  cos  A  ; 

or,  cos  A  =  (cos  a  —  cos  b  cos  c)  /sin  6  sin  c  ; 

which  expresses  a  dihedral  angle  in  terms  of  the  face  angles. 

6.  Express  a  face  angle  in  terms  of  the  dihedral  angles. 
(Employ  the  property  of  the  reciprocal  corner.) 

7.  If  the  face  angles  of  a  three-faced  corner  are  each  60°,  show 
that  the  cosine  of  a  dihedral  angle  is  t. 

8.  In  46  where  is  the  locus  if  A,  B,  C  are  in  line  ? 

9.  In  47  where  is  the  point  0  if  the  four  points  be  complanar  ? 
where  if  three  points  be  in  line  ? 


SECTION  4. 

POLYHEDRA. 

47.  Def.  A  spatial  figure  formed  of  four  or  more 
planes  so  disposed  as  to  completely  enclose  a  portion  of 
space  is  a  polyhedron.  It  is  analogous  to  the  polygon  in 
plane  geometry,  and  its  plane  section  is  always  some 
form  of  polygon. 

The  faces  of  the  polyhedron  are  those  portions  of 
planes  which  are  concerned  in  forming  the  closed  figure, 
but  for  generality  the  term  is  sometimes  extended  to 
outlying  parts  of  these  planes. 

The  adjacent  faces  meet  by  twos  to  form  edges,  and 
the  edges  are  concurrent  in  groups  of  three  or  more  to 
form  corners. 

When  a  polyhedron  is  such  that  no  line  can  meet  more 
than  two  of  its  faces,  it  is  convex. 

48.  Theorem.  In  any  polyhedron  the  sum  of  the  num- 
ber of  faces  and  the  number  of  corners  is  greater  by  two 
than  the  number  of  edges. 

Proof.  Any  polyhedron  may  be  supposed  to  be  built 
up  by  beginning  with  one  face,  and  to  it  adding  a  second 
face,  and  then  a  third,  and  so  on  until  the  figure  is 
completed. 

Denote,  in  general,  the  number  of  corners  by  C,  the 
number  of  faces  by  F,  and  the  number  of  edges  by  E. 

45 


46- 


SOLID   OR   SPATIAL   GEOMETRY. 


1.  Let  us  start  with  a  single  face,  U.  The  number  of 
edges  is  the  same  as  the  number  of  corners,  and  we  have 
one  face.    Therefore  the  equation  C-\-F  =  E+l\?,  satisfied. 

2.  To  C/add  the  face  V.     In  so  doing 

V  loses  one  of  its  edges,  BG,  and  two 
of  its  corners,  B  and  C,  by  union  with 
similar  parts  of  U.     So  that  in  adding 

V  we  increase  F  by  1,  and  we  increase 
E  by  one  more  than  the  increase  of  C; 
and  hence  the  equation  G-\-F=  E -\-l 
is  still  satisfied. 

3.  To  U  and  V  add  W.  This  new  face  loses  two  of 
its  edges,  DC  and  CG,  and  three  of  its  corners,  D,  G,  and 
O.  Here  again  we  add  one  face  and  one  more  edge  than 
corner,  so  that  G-\-F=E-{-Hb  still  satisfied. 

4.  It  is  readily  seen  that  in  adding  any  face  whatever, 
that  face  loses  one  more  corner  than  edge  by  union  with 
other  faces,  until  we  come  to  the  last  face  necessary  to 
complete  the  polyhedron. 

This  face  loses  all  its  edges  and  all  its  corners,  so  that 
by  adding  this  face  we  increase  the  number  of  faces  Vy 
1  without  interfering  with  the  numbers  of  edges  or  cor- 
ners.    And  hence  in  the  completed  polyhedron  we  have 

G+F=E+2. 

This  beautiful  theorem  is  usually  attributed  to  Euler, 
and  is  known  as  Euler's  theorem  on  Polyhedra,  but  it 
appears  to  have  been  known  before  his  time. 


THE   TETRAHEDRON. 


47 


Classification  of  Polyhedra. 

49.  Polyhedra  may  be  classified  as  follows : 

1.  Tetrahedron. 

2.  Parallelepiped,  Cuboid,  Cube. 

3.  Pyramid,  Frustum  of  Pyramid. 

4.  Prism,  Truncated  Prism. 

5.  Prismatoid,  Prismoid. 

6.  The  five  Kegular  Polyhedra. 

7.  A  number  of  Semi-regular  Derived  Polyhedra. 

This  classification  is  not  exhaustive,  and  its  divisions 
are  not  mutually  exclusive.  It  includes,  however,  all  the 
polyhedra  usually  met  with. 

Polyhedra  are  not  equally  important  in  any  sense,  and 
only  a  few  can  be  said  to  be  important  in  a  descriptive 
sense. 

The  Tetrahedron. 

50.  The  three  planes  which  form  a  three-faced  corner, 
and  any  fourth  plane,  not  through  the  vertex,  which  cuts 
them  all,  form  the  closed  figure 
called  a  Tetrahedron. 

The  tetrahedron  ^  BOD  has 
four  triangular  faces,  four 
three-faced  corners,  and  hence 
four  Vertices  and  six  edges, 
i.e.  the  six  joins  of  four  non- 
complanar  points  no  three  of 
which  are  in  line. 


Def.   Any  face  of  the  tetra- 
hedron may  be  taken  as  the  base  of  the  figure. 


The 


48        SOLID  OR  SPATIAL  GEOMETRY. 

three  edges  which  bound  the  base  are  then  called  basal 
edges,  and  the  other  three  are  lateral  edges. 

The  joins  of  the  middle  points  of  opposite  edges  are 
diameters.  There  are  thus  three  diameters,  EG,  FH, 
and  IJ. 

51.  Theorem.  The  diameters  of  a  tetrahedron  bisect 
one  another. 

Proof.  ABCD  is  a  skew  quadrilateral,  and  BD  and 
AC  are  its  diagonals. 

But  the  joins  of  the  middle  points  of  opposite  sides  of 
a  skew  quadrilateral  bisect  one  another  (30). 

Therefore  EG,  FH,  and  IJ  bisect  oneTnbther. 

Def.  1.  The  point  of  concurrence  of  the  diameters  is 
the  centre  of  the  tetrahedron.  And  a  section  through  the 
centre  parallel  to  a  pair  of  opposite  edges  is  a  middle 
section,  as  EFGH. 

Cor.  There  are  three  middle  sections,  and  these  pass 
through  the  middle  points  of  the  six  edges  taken  in 
groups  of  four. 

The  middle  sections  are  evidently  parallelograms,  and 
they  intersect  by  twos  along  the  three  diameters. 

Def.  2.  A  median  of  a  tetrahedron  is  the  join  of  a 
vertex  with  the  centroid  (P.  Art.  85.  Def.  2)  of  the 
opposite  face. 

There  are  thus  four  medians,  one  to  each  face. 

52.  Theorem.  The  medians  of  a  tetrahedron  pass 
through  the  centre,  and  are  divided  at  that  point  so  that 
the  part  lying  between  the  centre  and  a  face  is  one- 
fourth  of  the  whole  median  to  that  face. 


THE  TETRAHEDRON. 


49 


In  the  tetrahedron  ABCD,  I  is  the  middle  point  of 
'BC,  and  J  of  AD,  and  IJ  is  thus 
a  diameter. 

IP  is  one-third  of  lA,  and  P  is 
thus  the  centroid  of  the  face 
ABC  (P.  Art.  85),  and  DP  is 
the  median  to  the  face  ABC. 

Evidently  DP  and  IJ  are  com- 
planar,  and  intersect  in  some 
point  0.     Then  0  is  the  centre. 

J'roof.     Draw  JQ  II  to  DP  to  meet  lA  in  Q. 
Then,  as  J  is  the  middle  point  of  AD,  so  Q  is  the 
middle  point  of  AP  (P.  Art.  84.  Cor.  2). 

And  •.•  QP=PI  and  OP  is  II  to  JQ,  0  is  the  middle 
point  of  JI,  and  is  therefore  the  centre  (P.  Art.  84. 
Cor.  2).     Hence  the  medians  pass  through  the  centre. 

Again,  PO  =  l  QJ,  and  QJ=  ^  PD. 

.-.  PO  =  \PD. 


Note.  —  A  face  of  a  polyhedron  is  a  segment  of  a  plane,  and  is 
in  form  triangular,  rectangular,  etc.  But  in  order  to  avoid  such 
uncouth  words  as  parallelogramic  we  shall  speak  of  these  faces  as 
being  triangles,  rectangles,  parallelograms,  etc.,  although  not  using 
these  terms  strictly  as  defined  in  plane  geometry.  This  usage  will 
shorten  language  and  cannot  possibly  lead  to  confusion. 


60  SOLID   OR   SPATIAL   GEOMETllY. 

The  Parallelepiped. 

53.  Def.  The  parallelepiped  has  six  faces,  of  which 
each  pair  of  opposite  ones  are  parallel  planes. 

The  contraction  ppd.  will  be  frequently  used  for  the 
word  ^parallelepiped.'  • 

Since  parallel  planes  cut  any  other  plane  in  parallel 
lines  (Art.  21),  and  since  the  planes  AC  and  A'O  are 
parallel  and  cut  the  paral- 
lel planes  AD'  and  A'D,  it 
follows  that  AB,  CD,  A'B', 
and  CD'  are  all  parallel. 
Similarly,  AD,  BC,  A'D', 
and  B'C  are  parallel,  and 
AC,  A'C,  BD',  and  B'D 
are  parallel. 

Thus  the  faces  of  a  ppd. 
are  parallelograms  congruent  in  opposite  pairs,  and  the 
twelve  edges  are  in  parallel  sets  of  four  in  each  set. 

The  corners,  which  are  eight  in  number,  are  each 
three-faced,  and  the  three  edges  which  meet  at  any  one 
vertex  give  the  directions  of  all  the  edges,  and  these  are 
therefore  called  direction  edges. 

Cor.l.  As  the  Z  BAD  =  Z  B'A'D',  the  Z  DAC  = 
Z  D'A'C,  and  the  ZBAC  =  Z  B'A'C,  the  corners  having 
their  vertices  at  A  and  A'  contain  face  angles  which  are 
respectively  equal,  but  these  are  disposed  in  opposite 
orders  about  the  vertices. 

The  same  is  true  for  any  other  pair  of  opposite  cor- 
ners. 


THE   PARALLELEPIPED.  51 

Therefore,  opposite  corners  of  a  parallelepiped  are 
symmetrical. 

Cor.  Considering  three-faced  corners  composed  of  the 
same  face  angles  as  being  of  the  same  variety,  there  are 
at  most  only  four  varieties  of  corner  in  any  parallelepiped. 

These  will  be  called  representative  corners. 

54.  By  considering  the  forms  of  representative  corners, 
all  ppds.  may  be  divided  into  two  classes,  the  acute  and 
the  obtuse. 

A  denoting  any  angle,  let  A'  denote  its  supplement. 

Let  A,  B,  G,  all  acute  or  all  obtuse,  be  the  three  face 
angles  at  one  corner  of  a  ppd. 

Then  the  representative  cor- 
ners are  easily  seen  to  be  ABC, 
AB'C,  A'BC,  and  A'B'C. 

(1)  If  A,  B,  C  are  acute, 
A',  B',  C  are  obtuse. 

Therefore,  if  a  parallelepiped 
has  one  corner  formed  of  acute  face  angles,  the  other  rep- 
resentative corners  contain   one   acute   and  two   obtuse 
face  angles,  each. 

This  is  an  acute  parallelepiped. 

(2)  If  A,  B,  C  are  obtuse,  A',  B',  C  are  acute. 
Therefore,  if  a  parallelepiped  has  one  representative 

corner  composed  of  obtuse  face  angles,  the  other  repre- 
sentative corners  have,  each,  one  obtuse  and  two  acute 
face  angles. 

This  is  an  obtuse  parallelepiped. 

It  thus  appears  that  no  one  ppd.  can  contain  all  the 
kinds  of  corners  belonging  to  ppds. 


62  SOLID   OR   SPATIAL   GEOMETRY. 

55.  Def.  The  join  of  opposite  vertices  in  a  ppd.  is 
a  diagonal.  These  are  four  in  number,  viz.  AA\  BB', 
CC,  and  DD'  (Fig.  of  53). 

Since  AD  is  II  to  BC,  is  II  to  D'A'  and  equal  to  it, 
AD'A'D  is  a  parallelogram,  and  its  diagonals  bisect  one 
another.  Hence  AA'  and  DD'  bisect  one  another ;  and 
similarly,  AA'  and  BB'  bisect  one  another,  etc. 

Therefore,  all  the  diagonals  of  a  ppd.  pass  through 
a  common  point,  and  are  bisected  at  that  point. 

The  common  point  of  the  diagonals  is  the  centre. 

56.  Theorem.  Every  line-segment  passing  through 
the  centre  of  a  parallelepiped,  and  having  its  end-points 
upon  the  figure,  is  bisected  at  the  centre. 

Proof.  PQ  (Fig.  53)  is  a  line-segment  passing  through 
the  centre,  0,  and  having  its  end-points  P,  Q  in  the  face 
AC  and  A'C  respectively. 

Join  AP  and  A'Q.  Then  AP  and  A'Q  are  complanar, 
since  PQ  passes  through  0;  and  the  plane  of  ^Pand 
A'Q  cuts  the  parallel  faces  AC  and  A'C  in  parallel  lines 
(Art.  21.  Cor.  1). 

.-.  ^Pis  II  to  A'Q. 

Also,       AO  =  A'0,  and  ZAOP=ZA'OQ, 

and  ZOAP=ZOA'Q. 

.:  AAOP  =  AA'OQ, 

and  OP=OQ. 

Cor.  The  centre  of  a  ppd.  is  the  centre  of  every  cen- 
tral section. 


THE   PARALLELEPIPED.  53 

57.  As  a  parallelepiped  has  three  direction  edges,  three 
sections  may  be  made  normal  to  each  of  these  edges 
respectively.  These  sections  will  be  forms  of  the  paral- 
lelogram, 

Def.  1.  If  none  of  the  sections  are  rectangles,  the 
ppd.  is  tricUnic,  and  none  of  its  angles,  whether  face  or 
dihedral,  are  right  angles. 

2.  If  one  section  is  a  rectangle,  the  ppd.  is  diclinic, 
and  four  dihedral  angles,  whose  edges  are  parallel,  are 
right  angles. 

3.  If  two  sections  are  rectangles,  the  ppd.  is  mono- 
clinic,  and  two  sets  of  four  dihedral  angles  are  right 
angles. 

4.  If  the  three  sections  are  rectangles,  all  the  faces 
are  rectangles,  and  all  the  dihedral  angles  are  right 
angles,  and  all  the  corners  are  right  corners  (Art.  40. 
Def.).     The  figure  is  then  a  cuboid.^ 

Cor.  In  the  cuboid  all  the  diagonals  are  equal,  and 
the  direction  lines  are  mutually  perpendicular  to  one 
another. 

Def.  2.  A  cuboid  with  its  edges  equal  is  a  cube.  The 
faces  of  the  cube  are  squares. 

The  analogues  of  the  ppd.,  the  cuboid,  and  the  cube, 
are  in  plane  geometry  the  parallelogram,  the  rectangle, 
and  the  square. 

1  This  term  was  proposed  by  Mr.  Hay  ward.  Before  the  appearance 
of  Mr.  Hayward's  work  I  used  the  term  orthopiped  for  a  rectangular 
parallelepiped.  But  cuboid  is  evidently  a  better  and  a  more  convenient 
term. 


64  solid  or  spatial  geometry. 

The  Pyramid. 

58.  Def.  1.  When  a  corner  of  any  number  of  faces 
is  cut  by  a  plane  which  cuts  all  the  faces,  the  closed 
figure  so  formed  is  called  a  pyramid. 

The  cutting  plane  is  the  base,  and  the  planes  which 
form  the  corner  are  faces  of  the  pyramid.  The  edges 
which  bound  the  base  are  hasal  edges,  and  those  which 
belong  to  the  corner  are  lateral  edges.  The  vertex  of  the 
corner  is  the  vertex  or  apex  of  the  pyramid. 

Def.  2,  Pyramids  are  classified  into  triangular,  square, 
etc.,  according  to  the  character  of  the  base.  A  triangular 
pyramid  is  a  tetrahedron. 

59.  Def.  If  a  pyramid  be  cut  by  a  plane  parallel  to 
its  base,  the  portion  lying  between  the  base  and  this  cut- 
ting plane  is  called  2i  frustum  of  a  pyramid. 

The  frustum  has  thus  two  bases,  a  lower  and  an  upper, 
or  a  major  base  and  a  minor  base. 

From  Art.  28.  Cor.  2,  it  follows  that  the  two  bases  of 
the  frustum  of  a  pyramid  are  similar  polygons. 

The  Prism. 

60.  When  the  vertex  of  a  pyramid  goes  to  infinity  in 
a  direction  normal  to  the  base,  the  lateral  edges  become 
parallel  lines,  and  the  resulting  figure  is  not  a  closed 
figure.  But  under  like  circumstances  the  frustum  be- 
comes a  closed  figure  with  two  congruent  bases,  and  is 
called  a  prism. 

If  one  edge  of  a  prism  is  normal  to  a  base,  all  the 
edges  are  normal,  and  the  lateral  faces  are  rectangles. 
This  is  called  a  right  prism. 


THE   REGULAR   POLYHEDRA.  55 

And  if  one  of  the  lateral  edges  is  inclined  to  the  base, 
they  are  all  inclined  at  the  same  angle.  This  is  an 
oblique  prism. 

Prisms  are  usually  named  from  the  character  of  the 
right  section.  Thus  a  right  rectangular  prism  is  a  cuboid, 
and  a  parallelepiped  may  be  a  right  prism  or  an  oblique 
prism,  depending  upon  its  kind  (Art.  57). 

The  Regular  Polyhedra. 

61.  Def.  A  regular  polyhedron  is  one  in  which  all 
the  faces  are  regular  polygons  of  the  same  number  of 
sides,  and  all  the  corners  are  formed  by  the  same  number 
of  faces. 

This  implies  that  all  the  edges  are  equal,  that  all  the 
face-angles  are  equal,  and  that  all  the  dihedral  angles 
are  equal. 

On  account  of  the  perfect  symmetry  of  the  figure,  it 
must  have  a  definite  centre  equally  distant  from  each 
face  and  equally  distant  from  each  vertex.  The  normal 
at  the  centre  of  each  face  passes  through  the  centre  of 
the  figure,  and  the  line  from  a  vertex  to  the  centre  is  an 
isoclinal  to  the  edges  of  that  vertex  and  to  the  faces  of 
that  vertex. 

One  of  the  regular  polyhedra  is  familiarly  known  as 
the  cube. 

62.  Theorem.  There  cannot  be  more  than  five  regular 
polyhedra. 

Proof.  The  least  number  of  faces  which  can  form  a 
corner  is  three,  and  these  must  not  be  complanar.  There- 
fore the  three  face-angles  must  together  be  less  than  a 


56  SOLID   OR   SPATIAL  GEOMETRY. 

circumangle,  or  a  face-angle  must  be  less  than  four- 
thirds  of  a  right  angle  (Art.  42). 

The  only  regular  polygons  having  their  internal  angles 
less  than  |  of  a  right  angle  are  (P.  Art.  133.  Cor.)  the 
equilateral  triangle,  the  square,  and  the  regular  penta- 
gon; and  these  alone  can  form  the  face  of  a  regular 
polyhedron. 

Equilateral  Triangle. 

A  corner  may  be  formed  of  3,  4,  or  5  equilateral  tri- 
angles, and  may  therefore  be  three-,  four-,  or  five-faced. 

1.  The  three-faced  corner  gives  the  regular  tetrahedron, 
with  4  faces,  4  corners,  and  6  edges. 

2.  The  four-faced  corner  gives  the  regular  octahedron, 
with  8  faces,  6  corners,  and  12  edges. 

3.  The  five-faced  corner  gives  the  regular  icosahedron, 
with  20  faces,  12  corners,  and  30  edges. 

Square. 

Only  one  corner,  a  three-faced,  can  be  formed  by  squares. 

4.  This  gives  the  cube,  with  6  faces,  8  corners,  and  12 

edges. 

Regular  Pentagon. 

Only  one  corner,  a  three-faced  one,  can  be  formed. 

5.  This  gives  the  regular  dodecahedron,  with  12  faces, 
20  corners,  and  30  edges. 

These  are  the  five  regular  polyhedra. 

63.   Euler's  theorem,  Art.  48,  gives 

F+C=E  +  2. 

Now  the  numbers  denoted  by  F  and  C  are  evidently 
interchangeable,  while  E  remains  the  same.    That  is, 


THE  REGULAR   POLYHEDRA.  -      67 

if  we  have  a  given  polyhedron,  we  can  form  another 
polyhedron  in  which  the  number  of  corners  is  the  same 
as  the  number  of  faces  in  the  given  polyhedron,  and  the 
number  of  faces  is  the  same  as  that  of  the  corners  in 
the  first  polyhedron,  while  the  number  of  edges  remains 
the  same  in  both. 

These  polyhedra  may  be  called  reciprocals  of  each 
other,  as  either  may  be  formed  from  the  other  by  a  sort 
of  reciprocation,  the  changing  of  points  into  planes,  and 
planes  into  points. 

If  a  point  be  taken  in  each  face  of  any  polyhedron, 
preferably  the  centre  where  there  is  one,  and  these 
points  be  joined  in  every  way,  provided  we  join  only 
points  which  lie  on  adjacent  faces,  the  joins  form  the 
edges  of  a  polyhedron  which  is  reciprocal  to  the  original 
polyhedron. 

If  the  new  polyhedron  be  treated  in  the  same  way, 
we  obtain  a  third  polyhedron,  which  is  reciprocal  to  the 
second,  and  is  accordingly  of  the  same  species  as  the 
first. 

64.  Applying  the  principles  of  the  preceding  article 
to  the  regular  polyhedra,  we  readily  see  that  the  octa- 
hedron is  the  reciprocal  of  the  cube,  and  the  dodeca- 
hedron is  the  reciprocal  of  the  icosahedron. 

The  tetrahedron,  having  the  number  of  its  faces  and 
vertices  the  same,  gives  another  tetrahedron  by  recipro- 
cation ;  or  the  tetrahedron  is  self-reciprocal. 

65.  Interesting  models  of  all  the  polyhedra  may  be 
made  by  drawing  proper  figures  on  cardboard;  then 
cutting  out  the  entire  piece,  and  cutting  half-way  through 


58 


SOLID   OR   SPATIAL   GEOMETRY. 


the  remaining  lines.  The  piece  of  cardboard  may  now 
be  folded  along  these  lines  to  form  the  intended  figure, 
and  the  edges  be  fastened  together  with  glue. 

The  figure  drawn  on  the  cardboard  is  called  a  net. 

The  net  for  an  obtuse  parallelepiped  is  given  in  the 
diagram.     The  faces  are  denoted  by  U,  V,  and  W,  those 


/ 

a 

/ 

/ 

/e 

u 

^( 

/^ 

^/\ 

b' 

\      w" — 

/w     \ 

V 

\    i^ 

(7       ^ 

\ 

Xy^ 

\ 

^~— ~- 

/ 

/ 

4 

\ 

\ 

u 

X 

/ 

r 

1 

1 

/' 

V 

c 

having  the  same  letter  being  opposite,  and  therefore  con- 
gruent parallelograms.  The  edges  which  come  together 
are  denoted  by  the  same  small  letter.  Those  having 
the  same  letter  attached  must,  of  course,  be  the  same 
in  length.  The  three  obtuse  angles  concerned  are 
denoted  by  A,  B,  and  G.  All  the  other  angles  are  then 
known. 

If  the  angle  C  were  acute,  as  indicated  by  the  dotted 
lines,  the  ppd.  would  be  acute.  And  the  same  results 
would  be  obtained  by  making  either  Aov  B  acute. 

As  the  net  is  drawn,  the  ppd.  will  be  triclinic.  If  the 
U  faces  be  rectangles,  the  ppd.  will  be  diclinic ;  if  both 
U  and  V  are  rectangles,  it  will  be  monoclinic ;  and  if  all 


NETS. 


59 


the  faces  be  rectangles,  the  figure  will  be  the  cuboid ; 
and  if  all  squares,  the  cube. 

The  accompanying  diagrams  give  nets  for  the  regular 
polyhedra  other  than  the  cube. 


Nets   for  prisms   and  pyramids  and  frusta  need  no 
description. 


60  SOLID   OR   SPATIAL  GEOMETRY. 


EXERCISES  D. 

1.  The  faces  of  a  polyhedron  are  3  squares  and  2  triangles. 
Find  the  number  of  edges  and  of  corners  and  classify  the  figure. 

2.  If  an  n-hedron  has  all  its  faces  triangles,  the  number  of  its 
corners  is  J(n  +  4). 

3.  If  P,  Q,  i?,  S  be  the  centroids  of  a  tetrahedron,  the  recipro- 
cal having  P,  Q,  B,  S  as  vertices  has  the  same  centre  as  the  original. 
Also  the  diameters  and  medians  of  the  two  tetrahedra  coincide, 
except  in  length. 

4.  In  the  regular  tetrahedron  the  diameters  are  perpendicular 
to  one  another. 

6.  If  the  diameters  of  a  tetrahedron  tenninate  in  the  centres 
of  the  faces  of  a  cube,  then  the  edges  are  diagonals  of  the  faces. 
Thence  show  how  the  cube  may  be  transformed  into  a  regular 
tetrahedron. 

6.  If  AA',  BB',  CC,  and  DD'  are  diagonals  of  a  cuboid,  show 
that  the  middle  points  of  AB,  BC,  CA',  A'B',  B'C,  and  C'A  are 
complanar. 

Find  the  form  of  the  section  through  these  points. 

7.  The  join  of  A'  with  the  middle  point  of  AB,  and  the  join  of 
C  with  the  middle  point  of  BC,  divide  each  other  into  parts 
which  are  as  2  to  1  (Ex.  6). 

8.  The  centres  of  the  adjacent  faces  of  a  ppd.  are  joined.  What 
closed  figure  is  formed  ?    Describe  its  characteristics. 


SECTION   5. 

The  Cone,  the  Cylinder,  and  the  Sphere. 

66.  The  three  figures  here  mentioned  are  the  simplest 
spatial  figures  having  curved  surfaces,  and  they  are  fre- 
quently spoken  of  as  the  three  round  bodies. 

The  cone  and  the  cylinder  can  be  generated  by  the 
motion  of  a  straight  line,  and  they  are  consequently 
called  ruled  surfaces. 

The  sphere  is  not  a  ruled  surface,  but  a  surface  of 
double  curvature. 

Def.  A  surface  which  can  be  generated  by  the  revolu- 
tion of  a  plane  figure  about  an  axial  line  in  its  plane,  is 
a  surface  of  revolution. 

The  sphere  is  a  surface  of  revolution. 

The  cone  and  the  cylinder  may  or  may  not  be  surfaces 
of  revolution. 

Solid  geometry  furnishes  other  interesting  examples 
of  ruled  surfaces  besides  the  cone  and  cylinder,  and  of 
surfaces  of  revolution  besides  the  sphere.  As  examples 
of  the  first  we  have  the  common  conoid,  the  hyperboloid 
of  one  sheet,  and  the  elliptic  paraboloid;  and  of  the 
second,  the  oblate  spheroid,  the  prolate  spheroid,  and  the 
anchor  ring. 

The  Cone. 

67.  Def.  1.  In  general,  a  variable  line  which  passes 
through  a  fixed  point  and  is  guided  by  a  fixed  plane 

61 


62 


SOLID   OR   SPATIAL  GEOMETRY. 


curve,  not  complanar  with  the  point,  generates  a  cone,  or 
has  a  cone  as  its  locus. 

0  is  a  fixed  point,  and  APB  is  a  fixed  curve  not  com- 
planar with  the  point.      The 
variable  line  L  passes  through 
0,  and  meets  the  curve  APB. 
Then  L  generates  a  cone. 

Cor.  Since  L  is  unlimited 
in  length,  the  cone  extends  in- 
definitely outwards  upon  both 
sides  of  0,  and  is  not  a  closed 
figure. 

Def.  2.  0  is  the  centre  of 
the  cone,  and  the  two  parts 
into  which  it  divides  the  cone 
are  called  the  two  nappes  or 
sheets  of  the  cone. 

The  fixed  curve  APB  is  the  director,  and  the  line  L  is 
the  generator  of  the  cone. 

Any  line  which  coincides  with  the  generator  in  any  of 
its  positions  is  called  a  generating  line. 

Thus  every  line  passing  through  0  and  lying  on  the 
conical  surface  is  a  generating  line. 


68.  The  director  may  be  any  form  of  curve.  If  it 
becomes  a  line,  the  cone  degrades  into  a  plane  (Art.  7.  3) ; 
and  if  the  director  becomes  a  point,  the  cone  becomes  the 
line  through  that  point  and  the  centre. 

Thus  the  line  and  the  plane  may  be  looked  upon  as 
limiting  forms  of  the  cone. 


THE  CONE.  63 

69.  When  the  director  is  a  circle,  and  the  centre  0  is 
a  vertex  to  that  circle  as  a  cone-circle  (Art.  10.  Def.  1), 
the  cone  is  a  right  circular  cone,  and  the  line  through 
the  centre  of  the  circle  and  the  centre  of  the  cone  is  the 
axis  of  the  cone. 

The  circular  cone  is  a  figure  of  revolution,  and  is  the 
most  important  of  all  cones. 

The  word  'cone'  as  hereafter  employed  will  mean  a 
right  circular  cone,  unless  otherwise  qualified. 

70.  Let  C  (Fig.  of  67)  be  the  centre  of  the  circular 
director  APB.  Then  CP  is  constant,  and  GO  is  con- 
stant, and  OOP  is  a  1 .  Therefore  the  Z  POC  is  constant. 
This  angle  is  the  semi-vertical  angle  of  the  cone. 

Hence  a  circular  cone  is  generated  by  a  line  which 
revolves  about  a  fixed  axial  line  while  meeting  the  latter 
in  a  fixed  point  and  at  a  fixed  angle. 

Cor.  1.  Every  section  of  a  circular  cone,  normal  to 
the  axis,  is  a  circle. 

Cor.  2.  Every  section  of  a  circular  cone,  through  the 
axis,  is  two  lines  intersecting  at  a  fixed  angle  the  vertical 
angle  of  the  cone. 

Cor.  3.  Every  section  of  a  circular  cone  through  the 
centre  is  two  lines ;  for  the  plane  meets  the  cone  along 
two  generating  lines. 

Cor.  4.  Any  point  on  the  axis  of  a  circular  cone  is 
equidistant  from  the  surface  on  all  sides,  and  the  axis  is 
thus  an  isoclinal  line  to  the  surface. 

71.  Theorem.  Only  two  generating  lines  of  a  cone  are 
complauar. 


64  SOLID   OR   SPATIAL   GEOMETRY. 

Proof.  Since  the  generating  lines  all  pass  through  0, 
any  two  of  them  are  complanar. 

Let  any  two  particular  generating  lines  meet  the 
director  circle  in  A  and  P.  The  plane  of  these  lines 
meets  the  plane  of  the  circle  in  a  line  (5),  and  as  a  line 
can  meet  a  circle  in  only  two  points  (P.  Art.  94),  the 
plane  of  OA  and  OP  has  only  two  points  coincident  with 
the  circle,  and  therefore  only  two  generating  lines  lie  in 
this  plane. 

72.  Theorem.  A  line  which  is  not  a  generating  line 
can  meet  a  cone  in  only  two  points. 

Proof.  Let  Mhe.  the  line,  not  passing  through  0;  and 
let  the  plane  U  pass  through  0,  and  contain  M.  If  U 
cuts  the  cone,  it  contains  two  generating  lines ;  and  since 
it  contains  M,  the  two  generating  lines  are  complanar 
with  M,  and  meet  it  in  two  points,  and  in  only  two 
points ;  and  these  points  are  common  to  M  and  to  the 
cone. 

Therefore,  the  line  M  can  meet  the  cone  in  two,  and  in 
only  two,  points. 

73.  If  the  two  points  in  which  a  line  M,  which  is  not 
a  generating  line,  meets  a  cone  become  coincident,  the 
line  becomes  a  tangent  line  to  the  cone,  and  has  one  point 
only,  a  double  point  (P.  Art.  109.  Def.  2)  in  common 
with  the  cone. 

The  plane  determined  by  a  tangent  line  and  the  generat- 
ing line  through  its  point  of  contact  is  a  tangent  plane 
to  the  cone,  and  touches  the  cone  along  this  generating 
line,  which,  as  it  represents  the  union  of  two  lines,  is  a 
double  line. 


THE  CYLINDER.  65 

Cor.  1.  Evidently  all  tangent  planes  to  a  cone  pass 
through  the  centre  and  form  a  sheaf  of  planes. 

Cor.  2.  All  tangent  planes  to  a  cone  intersect  one 
another  in  lines  which  pass  through  the  centre  and  form 
a  sheaf  of  lines. 

74.  Def.  A  line  through  the  centre  perpendicular  to 
a  generating  line  of  a  cone  generates  a  second  cone,  which 
is  the  reciprocal  of  the  first. 

When  the  vertical  angle  of  the  cone  is  a  right  angle, 
these  two  cones  become  coincident,  and  form  but  one 
cone. 

The  Cylinder. 

75.  When  the  centre  of  a  cone  goes  to  infinity  in  the 
direction  of  the  axis,  and  the  director  curve  remains 
finite,  the  cone  becomes  a  cylinder,  and  the  axis  of  the 
cone  becomes  the  axis  of  the  cylinder.     Hence: 

Def.  1.  A  cylinder  is  the  locus  of  a  line  which  keeps 
a  fixed  direction  and  meets  a  fixed  plane  curve  which  is 
not  complanar  with  the  line. 

Def.  2.  A  circular  cylinder  is  generated  by  one  of  a 
pair  of  parallel  lines  while  revolving  at  a  fixed  distance 
about  the  other  parallel  as  a  fixed  axial  line.  The  fixed 
line  is  the  axis  of  the  cylinder. 

Cor.  1.     The  cylinder,  as  defined,  is  not  a  closed  figure. 

Cor.  2.     A  line  can  meet  a  circular  cylinder  twice,  and 

only  twice. 

Cor.  3.  Sections  of  a  circular  cylinder  normal  to  the 
axis  are  equal  circles. 


SOLID   OR   SPATIAL   GEOMETRY. 


The  Sphere. 

76.  Def.  A  sphere  is  the  locus  of  a  semicircle  which 
revolves  about  its  limiting  centre  line  as  an  axial  line. 

BAD  is  a  semicircle,  and  AB 
is  its  limiting  diameter.  When 
ADB  revolves  about  AB  as  an 
axis,  the  semicircle  generates  a 
sphere  of  which  OD  is  a  radius. 

Cor.  1.  All  the  radii  of  a 
sphere  are  equal  to  one  another. 
Therefore, 

Def.  A  sphere  is  a  surface 
every  point  on  which  is  equi- 
distant from  a  fixed  point  within  called  the  centre. 

Cor.  2.  The  sphere  is  a  closed  figure,  so  that  to  pass 
from  without  the  sphere  to  within,  or  from  within  to 
without,  it  is  necessary  to  cross  the  surface. 

Cor.  3.  A  point  is  within  a  sphere,  on  the  sphere,  or 
without  it,  according  as  its  distance  from  the  centre  is 
less  than,  equal  to,  or  greater  than  the  radius  of  the 
sphere. 

Cor.  4.  Two  spheres  which  have  the  same  centre  and 
the  same  radius  coincide  in  all  their  parts  and  form 
virtually  but  one  sphere. 

77.  Theorem.  Every  plane  section  of  a  sphere  is  a 
circle. 

Let  DEP  be  the  plane  section  and  P  be  any  point  on 
it  (Fig.  of  76). 


THE   SPHERE.  67 

Then,  0  being  the  centre  of  the  sphere,  OP  is  constant, 
and  P  lies  in  the  plane  of  section. 

Therefore  (Art.  10.  Cor.)  the  section  is  a  circle. 

Def.  The  section  by  a  plane  through  the  centre  of 
the  sphere  is  the  largest  circle  producible,  and  is  called 
a  great  circle  of  the  sphere.  All  other  sections  are  small 
circles. 

Cor.  A  great  circle  of  a  sphere  has  its  centre  coinci- 
dent with  that  of  the  sphere ;  and  the  generating  semi- 
circle of  the  sphere  is  one-half  of  one  of  its  great 
circles. 

78.  Theorem.  A  line  can  meet  a  sphere  in  two,  and 
in  only  two,  points. 

Proof.  If  a  line  meets  a  sphere,  any  plane  containing 
the  line  gives  in  section  a  circle  cutting  the  line ;  and  as 
the  circle  cuts  the  line  twice,  and  twice  only,  so  a  line 
can  meet  the  sphere  in  two,  and  in  only  two,  points. 

Def.  A  line  which  meets  a  sphere  is  a  secant  line,  and 
the  part  within  the  sphere  is  a  chord, 

A  secant  through  the  centre  is  a  centre  line,  and  its 
chord  is  a  diameter. 

A  plane  which  cuts  a  sphere  is  a  secant  plane,  and 
when  it  passes  through  the  centre  it  is  a  diametral  plane. 

79.  Theorem.  The  join  of  the  centre  of  a  sphere  with 
the  middle  point  of  a  chord  is  perpendicular  to  the 
chord. 

Let  DE  be  a  chord  whose  middle  point  is  C  (Fig.  of 
76)  ;  then  OG  is  X  DE, 


68  SOLID   OR   SPATIAL   GEOMETRY. 

The  plane  of  0  and  DE  gives  in  section  a  circle  with 
DE  as  chord,  and  0  as  centre.  And  C  being  the  middle 
point  of  the  chord,  OC  is  ±  DE  (P.  Art.  96.  4). 

Cor.  1.  Diameters  of  the  same  small  circle  bisect  one 
another,  and  being  chords  of  the  sphere,  the  join  of  the 
centre  of  a  small  circle  with  the  centre  of  the  sphere  is 
normal  to  the  plane  of  the  small  circle,  i.e.  to  the  plane 
of  section. 

The  converse  of  this  is  evidently  true. 

Cor.  2.  Lines  through  the  centres  of  small  circles 
and  respectively  normal  to  their  planes  meet  at  the 
centre  of  the  sphere. 

Cor.  3.  The  plane  normal  to  any  chord  at  its  middle 
point  contains  the  centre  of  the  sphere. 

For  this  plane  is  the  right-bisector  plane  of  the  chord, 
and  therefore  contains  every  point  equidistant  from  the 
end  points  of  the  chord.  But  the  centre  of  the  sphere 
is  equidistant  from  the  end  points  of  the  chord. 

80.   Problem.     To  find  the  centre  of  a  given  sphere. 

1st  Solution.  Draw,  on  the  sphere,  two  small  circles 
whose  planes  are  not  parallel,  and  draw  normals  to  the 
planes  of  these  circles  at  their  centres. 

These  normals  meet  at  the  centre  of  the  sphere  (Art. 
79.  Cor.  2). 

2d  Solution.  Draw  any  three  non-parallel  chords  and 
their  right-bisector  planes. 

These  planes  have  the  centre  as  their  common  point 
(Art.  79.  Cor.  3). 

Cor.  1.  In  the  first  solution,  if  the  planes  of  the 
circles  are  parallel,  the  normals  also  are  parallel ;  and 


THE   SPHERE.  69 

as  they  pass  through  the  same  point,  the  centre  of  the 
sphere,  they  are  coincident  (P.  Art.  70.  Ax.). 

Therefore  the  centres  of  parallel  sections  of  a  sphere 
lie  upon  a  centre  line  normal  to  the  planes  of  section, 
and  are  therefore  collinear. 

Cor.  2.  In  the  second  solution,  if  the  chords  are 
parallel,  so  also  are  their  right-bisector  planes;  and  as 
these  planes  are  concurrent,  they  are  also  coincident. 

Therefore,  the  middle  points  of  parallel  chords  in  a 
sphere  are  complanar,  and  lie  upon  a  diametral  plane 
normal  to  the  chords. 

81.  When  the  two  points  in  which  a  line  meets  a 
sphere  become  coincident,  the  line  becomes  a  tangent 
line  to  the  sphere  and  touches  the  sphere  in  a  double 
point. 

Hence,  for  a  sphere  to  touch  a  given  line  at  a  given 
point  is  equivalent  to  two  conditions. 

82.  Theorem.  A  tangent  line  to  a  sphere  is  perpen- 
dicular to  the  radius  to  the  point  of  contact,  and  con- 
versely. 

Proof.  The  plane  determined  by  the  tangent  line  and 
the  radius  to  the  point  of  contact  gives  in  section  a  circle 
with  its  tangent  line  and  radius,  and  as  the  same  angle  is 
involved,  the  truth  of  the  theorem  follows  (P.  Art.  110). 

Def.  An  indefinite  number  of  perpendiculars  may  be 
drawn  to  a  radius  at  its  extremity ;  these  are  all  tangent 
lines,  and  they  all  lie  in  a  plane  to  which  the  radius  is 
normal. 

This  plane  is  a  tangent  plane  to  the  sphere. 


70  SOLID   OR   SPATIAL  GEOMETRY. 

Cor.  A  tangent  plane  is  normal  to  the  radius  to  the 
point  of  contact. 

83.  Theorem.  Through  any  four  non-complanar  points, 
of  which  no  three  are  in  line,  one,  and  only  one,  sphere 
can  pass. 

Proof.  It  is  shown  in  Art.  46.  Cor.  3,  that  one,  and 
only  one,  point  is  equidistant  from  four  given  non-com- 
planar  points,  no  three  of  which  are  in  line. 

If  this  point  be  taken  as  centre,  and  its  distance  from 
any  one  of  the  given  points  be  taken  as  radius,  the  sphere 
so  determined  passes  through  the  four  given  points. 

Cor.  1.  Four  non-complanar  points,  no  three  of  which 
are  in  line,  determine  one  sphere. 

Cor.  2.  Spheres  which  coincide  in  four  non-complanar 
points  coincide  altogether. 

Def.  Four  or  more  points  so  situated  that  a  sphere 
can  pass  through  them  are  conspheric,  and  when  these 
points  form  the  vertices  of  a  figure,  the  figure  is  inscribed 
in  the  sphere,  and  the  sphere  circumscribes  the  figure. 

When  a  sphere  has  all  the  sides  of  a  skew  polygon  as 
tangent  lines,  the  sphere  is  inscribed  to  the  polygon,  and 
the  polygon  is  circumscribed  to  the  sphere. 

With  a  polyhedron  it  is  different.  For  a  sphere  may 
have  the  edges  as  tangent  lines,  or  the  faces  as  tangent 
planes,  but  not  both. 

The  sphere  having  the  edges  as  tangent  lines  is  the 
tangent  sphere  to  the  edges,  and  the  one  having  the  faces 
as  tangent  planes  is  the  tangent  sphere  to  the  faces. 


THE   SPHERE.  71 

Cor.  3.  A  regular  polyhedron,  on  account  of  its  com- 
plete symmetry,  has  all  its  vertices  conspheric,  all  its 
edges  tangent  lines  to  a  sphere,  and  all  its  faces  tangent 
planes  to  a  sphere,  and  these  three  spheres  have  the  same 
centre. 

84.  The  vertices  of  a  skew  quadrilateral  are  neces- 
sarily conspheric ;  for  from  the  definition  (29)  they  are 
four  non-complanar  points,  no  three  of  which  are  in  line. 
.  Let  A,  B,  C,  D  be  the  vertices  taken  in  order,  and  let 
the  sides  AB,  BC,  CD,  and  DA  be  considered  as  lines 
of  indefinite  length. 

Cor.  1.  Let  A  and  B  become  coincident.  Then  AB 
becomes  a  tangent  line  to  the  sphere. 

Therefore,  a  sphere  can  touch  a  given  line  at  a  given 
point,  and  pass  through  any  two  other  points  whose  join 
is  not  complanar  with  the  given  line. 

Cor.  2.  Let  A,  B,  and  C  become  coincident.  Then 
the  lines  AB  and  BC  become  two  tangent  lines  inter- 
secting on  the  sphere  at  B,  and  these  determine  a  tan- 
gent plane. 

Therefore,  a  sphere  can  touch  a  plane  at  a  given  point 
and  pass  through  any  one  other  point  which  does  not  lie 
in  the  plane. 

As  four  points  properly  situated  are  necessary  to 
determine  a  sphere,  touching  a  plane  at  a  given  point  is 
equal  to  three  conditions,  and  the  point  of  contact  is 
thus  a  triple  point. 

Cor.  3.  Let  A  and  B  become  coincident  at  one  point, 
and  C  and  D  become  coincident  at  another.     Then  the 


72  SOLID   OR   SPATIAL   GEOMETRY. 

line  AB  becomes  a  tangent  line  at  one  point,  and  the 
line  CD  a  tangent  line  at  another,  and  these  two  tangents 
are  not  complanar. 

Therefore,  a  sphere  may  touch  two  non-complanar 
lines  at  any  two  given  points,  one  in  each  line, 

85.  Theorem.  The  figure  of  intersection  of  two  spheres 
is  a  circle,  and  the  common  centre  line  of  the  spheres 
passes  through  the  centre  of  the  circle  and  is  normal  to 
its  plane. 

Proof.  Let  0  and  0'  be  the  centres  of  the  spheres, 
and  P  be  a  point  on  their  figure  of 
intersection  FQR.  Then  OP,  and 
O'P,  and  00'  are  constant  for  all 
positions  of  P.  Therefore,  P  lies 
on  a  cone-circle  to  which  0  and  0' 
are  vertices,  and  hence  00'  passes 
through  the  centre,  C,  of  the  circle, 
and  is  normal  to  its  plane. 

Cor.  1.  OP  and  O'P  being  given,  CP  decreases  as  00' 
increases,  and  vice  versa.  When  OPO'  is  a  right  angle, 
the  tangent  planes  to  the  two  spheres  are  perpendicular 
to  one  another,  and  the  spheres  intersect  orthogonally. 

Cor.  2.  When  P  comes  to  C,  the  circle  PQR  becomes 
a  point  upon  the  line  00'. 

Therefore,  when  two  spheres  touch,  they  do  so  at  a 
single  point,  and  the  common  centre  line  passes  through 
the  point  of  contact. 

86.  APBR  is  a  sphere  with  0  as  centre  and  0'  any 
point  without  the  sphere.     O'P  is  a  tangent  line  from 


THE   SPHERE. 


73 


0',  touching  the  sphere  at  P.     PQR  is  the  small  circle 
through  P,  whose  plane  is  normal  to  00'. 

1.  00'  and  OP  are  constants,  and 
Z  OPO'  is  a  right  angle,  since  O'P 
is  a  tangent  (Art.  82).  Therefore 
O'P  is  constant,  and  P  always  lies 
on  the  small  circle  PQR,  which  is 
a  cone-circle  to  0  and  0'  as  vertices. 

Therefore  all  tangent  lines  from 
a  given  point  to  a  sphere  are  equal. 

2.  O'P  is  the  generator  of  a  cir- 
cular cone  which  touches  the  sphere 
along  the  small  circle  PQR,  and  0'  is  the  centre  or  ver- 
tex of  the  cone. 

Def.  The  cone  of  which  O'P  is  the  generator  is  the 
tangent  cone  for  the  point  0'. 

The  circle  PQR  is  the  circle  of  contact,  and  its  plane 
is  the  polar  plane  of  the  point  0'  with  respect  to  the 
sphere ;  and  the  point  0'  is  the  pole  of  the  plane. 

3.  When  0'  comes  to  A,  the  tangent  cone  and  the 
polar  plane  of  0'  unite  to  form  the  tangent  plane  at  A ; 
hence  a  tangent  plane  is  a  double  plane  representing  the 
limiting  form  of  the  tangent  cone,  and  the  limiting  posi- 
tion of  the  polar  plane  as  the  pole  comes  to  the  sphere. 

Evidently,  then,  a  tangent  plane  is  the  polar  plane  to 
its  point  of  contact. 

87.  Problem.  To  find  the  locus  of  a  point  equidistant 
from  three  planes,  no  two  of  which  are  parallel,  and 
which  do  not  form  an  axial  pencil ;  i.e.  from  three  planes 
which  form  a  sheaf. 


74  SOLID   OR   SPATIAL  GEOMETRY. 

Let  ABC,  ACD,  ADB  be  the  planes  having  A  as  their 
common  point. 

Let  the  internal  and  external  bisecting  planes  of  the 
dihedral  angle  whose  edge  is 
AB  be  denoted  by  ab  and  AB  A^\ 

respectively,  and  similarly  for  /,'  \    ^\^^ 

the  other  dihedral  angles.  / ;   \  ^\^^ 

Also,  let  Ai  denote  the  com-    L-i—-\---^^-----^^ 

mon  line  of  ab  and  ac.     Then,       *^  r\\\      '^'./^/'^ 
as  every  point  on  ab  is  equidis-  \  ^^^^^^^'^ / 

tant  from  the  planes  ^BC  and  \       /^  /a^ 

ABD,  and  every  point  on  ac  \   j / 

is  equidistant  from  the  planes  '\/ 

ACB  and  ACD,  every  point  ° 

on  the  intersection  of  ab  and  ac,  that  is,  on  Aj,  is  equi- 
distant from  the  three  given  planes. 

The  line  Ai  is  thus  inclined  to  all  the  planes  at  the 
same  angle,  and  it  will  be  called  the  internal  isoclinal 
line  to  the  planes. 

Again,  every  point  on  AB  is  equidistant  from  the 
planes  ABC  and  ABD,  and  every  point  on  AC  is  equi- 
distant from  the  planes  ACB  and  ACD. 

Therefore,  every  point  on  the  common  line  of  AB  and 
AG,  that  is  on  the  line  A^,  is  equidistant  from  the  three 
planes,  and  A^  is  an  external  isoclinal  line  to  the  planes. 

Similarly,  A,,  and  A^  are  external  isoclinals  to  the  same 
three  planes. 

Therefore,  the  required  locus  consists  of  the  four 
isoclinal  lines  to  the  planes.  These  isoclinals  pass 
through  A,  the  common  point,  and  form  the  centre 
locus  of  a  sphere  which  touches  the  three  concurrent 
planes. 


THE  SPHERE.  75 

Cor.  Each  isoclinal  line  is  the  common  line  to  three 
bisector  planes  which  form  an  axial  pencil,  viz. : 

Ai  of  ah,  ac,  ad;  A,,  of  ab,  AC,  AD  ; 

Ac  of  ac,  AB,  AD;  and  Aa  of  ad,  AB,  AC. 

88.  Problem.  To  find  the  centre  of  a  sphere  which 
shall  touch  four  planes  so  situated  as  to  form  a  tetra- 
hedron. 

Employing  the  notation  of  Art.  87,  we  have  four  iso- 
clinal lines  to  three  of  the  planes,  at  each  vertex  of  the 
tetrahedron,  or  16  in  all.  These  are  Ai,  Bi,  Ci,  Di  as 
internal  ones,  and  A^,  A„  A^,  -B„,  B„  B^,  C„,  Cj,  C^,  D^, 
Df,,  D^,  as  external  ones. 

Denote  the  planes  opposite  A,  B,  C,  D  by  a,  p,  y,  8, 
respectively. 

Then  Ai  is  the  locus  of  a  point  equidistant  from 
ft,  y,  8;  and  £,.  from  a,  y,  8.  Therefore,  a  point  equidis- 
tant from  y  and  8  lies  upon  both  At  and  JB,-,  and  hence 
these  lines  intersect,  and  Oj  and  Df  pass  through  the 
point  of  intersection. 

Hence  (1)  Af,  B^,  Ci,  Z>,-,  meet  to  give  one  point 
required. 

Similarly,  each  of  the  following  groups  of  four  lines 
gives  a  point  equidistant  from  the  four  planes: 

(2)  Ai,  B^,  C„,  A ;  (3)  JS,  A„  C„  A; 
(4)  Ci,  A„  B,,  A ;  and  (5)  A,  A„  B„  C,. 

Again,  Aj,  is  the  locus  of  a  point  equidistant  from 
ft,  y,  8,  and  B^  from  a,  y,  8. 

Therefore,  Aj,  and  B^  intersect  in  a  point  equidistant 
from  a,  ft,  y,  8,  and  6^  and  A  pass  through  this  point. 


76  SOLID   OR   SPATIAL   GEOMETRY. 

Hence  these  lines  meet  in  groups  of  four  to  give  three 
points  equidistant  from  the  four  planes  ;  namely, 

(6)  A,  B^,  C„  A;   (7)  A,  a,  B„  A;  (8)  A„  D„,  B,,  C,. 

Thus  eight  spheres,  in  all,  can  be  found,  each  of  which 
shall  touch  four  planes  so  situated  as  to  form  a  tetra- 
hedron. 

EXERCISES  E, 

1.  If  the  director,  figure  in  the  generation  of  a  cone  (61)  is  a 
polygon,  what  figure  is  formed  ? 

2.  Show  that  the  cone  is  a  limiting  case  of  an  ?i-faced  corner, 
and  explain  how. 

8.  If  the  radius  of  a  sphere  is  the  generator  of  a  circular  cone, 
the  figure  of  intersection  of  the  sphere  and  cone  is  a  circle. 

4.  The  centre  locus  of  a  sphere  which  touches  a  plane  at  a  given 
point  is  a  normal  to  the  plane  at  the  given  point. 

6.  What  is  the  centre  locus  of  a  sphere  which  touches  a  line  at 
a  given  point  ?  which  touches  two  parallel  lines  ?  which  touches 
two  intersecting  lines  ?   which  touches  two  intersecting  planes  ? 


Part  II. 

AREAL  RELATIONS  INVOLVING  LINE-SEGMENTS 
ABOUT  SPATIAL  FIGURES. 

89.  The  theorem  in  plane  geometry  that  the  square 
on  the  hypothenuse  of  a  right-angled  triangle  is  equal  to 
the  sum  of  the  squares  on  the  sides,  the  theorem  that 
the  rectangle  on  the  parts  of  a  secant  line  between  a 
point  and  a  circle  is  equal  to  the  square  on  the  tangent 
from  the  point  to  the  circle,  and  others  of  this  nature, 
express  areal  relations,  involving  line-segments  of  plane 
figures. 

Many  important  relations  of  a  similar  nature  exist 
among  the  line-segments  connected  with  spatial  figures. 
These  we  propose  to  consider  in  this  part  of  the  work. 

77 


SECTION  1. 


The  Skew  Quadrilateral  and  the 
Polyhedron. 

90.  Theorem.  In  a  skew  quadrilateral  the  sum  of  the 
squares  on  the  sides  is  greater  than  the  sum  of  the 
squares  on  the  diagonals,  by 
four  times  the  square  on  the 
join  of  the  middle  points  of 
the  diagonals. 

Proof.  ABCD  is  a  skew- 
quadrilateral,  and  AC  and 
BD  are  its  diagonals,  having 
7  and  J  as  their  middle 
points. 

i>7is  median  to  A  CD  A,  and  JB/is  median  to  A  CBA. 
.-.  CD''+DA^+CB''+BA^=2  {CP^DP+CP+BP); 
or,  'S,AB'  =  4.CP  +  2DP^2BP. 

But  IJ  is  median  to  the  A  DIB ; 

.-.  2DP  +  2BP  =  4.BP  +  A.IP; 
or,  %AB'  =  4.CP->r4.BP-\-4.IJ^ 

=  CA'  +  BD'-h4.IP.  Q.E.D. 

This  important  theorem  is  true  of  all  quadrilaterals, 
whether  plain  or  skew  (P.  Art.  173). 

78 


THE   SKEW  QUADRILATERAL.  79 

91.  For  the  tetrahedron  let  us  adopt  the  following 
notation :  Taking  ABC  as  the  base  and  D  as  the  vertex, 
denote  the  lateral  edges  DA,  DB,  DC  by  a,  h,  and  c 
respectively,  and  the  basal  edges  BC,  CA,  AB  by  a^, 
6j,  and  q  respectively.  Then  a  and  ai  are  opposite 
edges,  etc. 

Theorem.  In  any  tetrahedron  four  times  the  sum  of 
the  squares  on  the  diameters  is  equal  to  the  sum  of  the 
squares  on  the  edges. 

Proof.  The  skew  quadrilateral  with  its  diagonals 
forms  the  tetrahedron. 

The  results  of  Art.  90  give : 

4 IJ'  =  a'  +  c'  +  tti^  +  c/  -b'-  bi\  (Fig.  of  90. ) 

4.EG'  =  a2  +  &2  +  ai^  +  6/ _  c^ _  ^l 

Therefore,  by  addition, 

4  (7/2  +  FH'  +  EG')  =  a' ■}- b' -\- c' +  a/  +  b,'  +  c/ ; 

or,  denoting,  in  general,  a  diameter  by  d  and  an  edge 
bye, 

4  2d'  =  ^e". 

Cor.  In  the  regular  tetrahedron,  all  the  diameters 
being  equal,  and  all  the  edges  being  equal,  gives, 

M'  =  3d%a.ndi%e'  =  6e^; 

So  that  if  the  diameter  is  equal  to  the  side  of  a  square, 
the  edge  is  equal  to  the  diagonal  of  the  square  (F.  Art. 
180.  Cor.). 


80  SOLID  OR   SPATIAL   GEOMETRY. 

92.  Theorem.    In  any  tetrahedron,  nine  times  the  square 
on  a  median  is  equal  to  the  dif- 
ference  between  three  times  the  A 

sum   of  the   squares  on  the  con-  A''\\\ 

terminous  edges  and  the  sum  of  /  /  \\  \ 

j/    '    \  \    \ 
the    squares    on    the    remaining  A^-J    \\    \ 

edges.  /  '\  /  ^""v\  \      \ 

Proof.    In  the  tetrahedron   aZ--— V T    \'-/ 

D'ABC,   P   is  the    centroid    of       ^^^"^^-~-!.\/ 
ABC.     Then  DP  is  the  median  b 

to  the  face  ABC. 

Bisect  AP  in  Q.     And  AQ=  QP=  PI=^AL 

DQ  is  median  to  the  AADP; 

.:  AD'  +  DP'  =  2AQ'  +  2DQ\        (P.  Art.  173.) 

Also,  DP  is  median  in  the  A  QDI; 

.:  DQ'  +  DP  =  2QP'  +  2DP'. 

And  eliminating  DQ'  between  these  relations,  we 
obtain 

SDP'  =  AD'  -\-2DP-  ^AP. 

But  •••  -47  is  median  in  the  A  ABC,  and  DI  is  median 
inthe  A5CA 

.-.  2  AP  =  AB'  +  AC -2BP=  c,'  +  b^' -  ^a^', 
and  2DI'  =  DB'-{-DC'-2BP  =  b'  +  c'  -^a^'; 

whence    3  DP'  =  a'  +  b'  +  c'-\  {a,'  +  by'  -f  Cy') , 
or  9  DP'  =  3  2a2  -  Sail  q.  e.  d. 

93.  Theorem.  In  any  tetrahedron,  nine  times  the  sum 
of  the  squares  on  the  medians  is  equal  to  four  times  the 
sum  of  the  squares  on  the  edges. 


THE   SKEW    QUADRILATERAL.  81 

Proof.     Let  wii,  nio,  m^,  nii  denote  the  medians. 

Then  from  86, 

D  as  vertex,  9 m^'  =  3 a'  +  3 6^  +3c'  -  a^' -bi"  -  c,^ ; 

A  as  vertex,   9  mi  =  3  a"  +  3  ftj^  +  3  ^^  _  «/  -  6^  _  c^ ; 

B  as  vertex,    ^m^  =  3ai  +3h'^  +  3ci  —a-  —  by  —  c^; 

C  as  vertex,    9 m^^^  =  3 a^'  +  3bi' ^3c'  -  a'  -b'  -  c,'. 

Whence  by  addition,  and  denoting  a  median  in  general 
by  m  and  an  edge  in  general  by  e,  we  have 

9  2m'  =  42el  q.e.  d. 

Cor.  1.  In  the  regular  tetrahedron  9  2m^  =  36  m^,  and 
4  2e' =  246^5 

.-.  m^  =  |el 

Cor.  2.  The  median  in  a  regular  tetrahedron  is  the 
same  as  the  perpendicular  from  the  vertex  to  the  base, 
and  denoting  it  by  p,  we  have 

Cor.  3.  Denoting  a  dihedral  angle  of  the  regular 
tetrahedron  by  E, 

sin  E=DP/DI=le-y/6-i- ^6^3=1-^2. 
And  cos  E  =  ^. 

94.  In  the  regular  tetrahedron  we  have  the  circum- 
scribed sphere,  the  tangent  sphere  to  the  edges,  and  the 
inscribed  sphere.  Denoting  the  radii  of  these  by  M,  p, 
and  r  respectively. 


E=0D^ip  =  le-^6, 

(Art.  52.) 

p=OI  =:^d  =  \e^2, 

(Art.  91.  Cor.) 

r=OP=\p  =  ^e^Q; 

i?2 :  />2 :  9-2  =  9  :  3  : 1. 

82  solid  or  spatial  geometry. 

The  Parallelepiped. 

95.   Denote  the  direction  edges  by  a,  b,  c,  and  an  edge 
in  general  by  e,  and  a  diagonal  in  general  by  d. 

Theorem.  The  sum  of  the 
squares  on  the  diagonals 
is  equal  to  the  sum  of  the 
squares  on  the  edges. 

Proof.  Since  the  faces 
are  all  parallelograms,  and 
AB  is  II  to  B'A',  AC  to 
C'A',  etc., 

AA'^  +  BE"  =  AB"  +  BA"  +  A'B"  +  B'A\ 
Similarly, 

CC"  +  DD'-  =  CD'  -f  DC"  +  CD''  +  D'CK 
Whence,  by  addition, 
ScT'  =  AB'  +  CD'  +  A'B"  +  CD"  +  BA" 

+  D'C  +  B'A'  +  DC". 
And     BA"  +  CD"  =  BC  +  CA"  +  A'D"  +  D'B' ; 
and         B'A'  +  CD'  =  DA'  +  AC  +  CB"  +  S'D^. 

.-.  2d-  =  2e^  Q.E.D. 

Cor.  1.     As  the  edges  are  separable  into  three  groups 
of  four  equal  edges  each  (Art.  53), 

^d'  =  4.{a'-{-b'  +  c'). 
Cor.  2.     In  the  cuboid  the  diagonals  are  all  equal,  and 

cP=a'-{-b'-\-c'. 
Cor.  3.     In  the  cube  a  =  &  =  c  j 
.-.  d'  =  3e'. 


THE  PARALLELEPIPED. 


83 


96.  Problem.  To  find  a  diagonal  of  a  given  parallele- 
piped by  a  plane  construction. 

In  Fig.  (1),  let  AO'BC  be  the  given  ppd.,  and  AA'  be 
the  diagonal  required. 


Analysis.  Draw  A'P,  BQ,  CR,  perpendiculars  on  AO, 
produced  if  necessary. 

P,  Q,  R  are  the  projections,  on  AO,  of  A',  B,  and  C. 

The  projection  of  the  middle  point  of  OA'  is  the  same 
point  as  the  projection  of  the  middle  of  BC,  i.e.  it  is  the 
middle  point  of  RQ. 

.:  OP=OR+OQ. 

Construction.  In  Fig.  (2)  construct  the  faces  AB, 
AC,  and  OA',  disposed  as  in  the  figure. 

Draw  CR,  BQ  1.  on  AO,  produced  if  necessary. 


84  SOLID   OR   SPATIAL   GEOMETRY. 

Take  OP=OR+  OQ,  and  draw  PT  ±  to  AO. 
With  0  as  centre  and  OA'  as  radius,  describe  a  circle 
cutting  PT  in  T.     Join  AT. 

.     AT  '\%  the  required  diagonal. 

Proof.  OR  is  the  same  for  both  figures,  and  so  also 
is  OQ,  and  therefore  OP',  and  AO  being  the  same  in 
both,  AP  is  the  same  in  both. 

Also,  or  of  Fig.  (2)  is  made  equal  to  OA'  of  (1). 

.-.  A  OPToi  (2)  =AOP^'of  (l),andPr=P^'. 

Hence  A  APT  of  (2)  =  A  APA'  of  (1), 

and  ^r  of  (2)  =  AA'  of  (1). 

In  like  manner  any  other  diagonal  can  be  constructed. 

Cor.     Let  the  face  angles  about  the  vertex  A'  be  all 
acute,  and  the  figure  is  an  acute  ppd.  (Art.  54). 
Denote 

Z  BA'C  by  A,  Z  CA'O'  by  v,  and  Z  O'A'B  by  /*. 

Then,  Fig.  (2), 

Z  BOG,  =  \,  Z  COP=fi,  Z  BOP=v. 

Now, 

^T"    ^  ylP2  ^  prp2  ^   (^0  +  0P)='  +  0^'2  -  OP^ 

=  AO'-\-2A0-0P-^0A" 
=  a^  +  2 a  (OQ  +  OP)  +  0A'\ 


THE   PARALLELEPIPED.  85 

But, 

0A'^=b'  +  c-  +  2  be  cos  X,  (P.  Art.  217. ) 

OQ    =  &  cos  V,  OR  =  c  cos  fi. 
.'.  AT'  =  AA'^  =  a^  -I-  62  +  c^  -t-  2  6c  cos  \  +  2 ca  cos  fx 

+  2  ah  cos  V. 

And  this  expresses  the  square  on  the  longest  diagonal, 
i.e.  the  one  extending  between  the  vertices  having  three 
acute  face  angles. 

The  other  diagonals  are  given  by  making  two  angles 
obtuse  in  every  possible  way.     They  are : 

a^  +  &^  +  c^  +  2  &c  cos  A  —  2  ca  cos  fx.  —  2ah  cos  v, 

a?  -\-h'  ■\-  c-  —  2hc  cos  \-\-2ca  cos  fx  —  2ah  cos  v, 

a?  +  h'^  +  c-  —  2hc  cos  A  —  2  ca  cos  fi-\-2ab  cos  v. 

For  the  diagonals  of  an  obtuse  ppd.  it  is  only  neces- 
sary to  change  throughout  the  algebraic  sign  of  every 
cosine  term. 

Remark.  In  making  constructions  like  the  foregoing  care  must 
be  exercised  that  every  measured  segment  is  taken  in  its  proper 
sense,  or  with  its  proper  sign. 

By  taking  the  face  angles  about  A'  all  acute,  the  per- 
pendiculars A'P,  BQ,  CR,  all  fall  to  the  right  of  0. 
Under  a  different  arrangement  of  angles,  some  or  all 
of  these  might  have  fallen  to  the  left  of  0.  In  any  case 
if  M  is  the  middle  point  of  QR,  OP  is  to  be  taken  equal 
to  2  OM,  whatever  the  sign  may  be. 

97.  Let  O-ABO  be  a  cuboid,  and  OP  be  a  diagonal. 
Then  0P^=  OA' +  OB" -\- 00-  (Art.  95.  Cor.  2). 


86 


SOLID   OR   SPATIAL   GEOMETRY. 


But  if  OX,  0  Y,  OZ,  the  direction  lines  of  the  cuboid, 
meeting  in  0,  be  taken  as  the 
three  rectangular  axes  •  of  space 
(Art.  8.  Def.  1),  0^  is  the  pro- 
jection of  OP  on  OX,  OB  is  the 
projection  of  OP  on  OY,  and  OC, 
of  OP  on  OZ. 

Therefore,  the  square  on  any 
line-segment  is  equal  to  the  sum 
of  the  squares  of  the  projections 
of  the  segment  on  any  three 
mutually  perpendicular  lines. 

98.   Denoting  OA  by  a,  OB  by  b,  and  OC  by  c ;  also 
ZPOA  by  a,   Z  POB  by  /3,   Z  POC  by  y,  we  have 


.  -X. 


cos' a 


2/y  — 


OA' 


OP'     a^-i-  b'-\-  c^' 
with  the  symmetrical  expressions  for  cos^yS  and  cos^y. 
.-.  cos^a-f  cos^^ -f-cos^y  =  1. 

Def.  The  angles  a,  ^,  y  are  direction  angles  of  the 
line  OP,  and  determine  the  direction  of  OP  relatively  to 
the  three  axes.  The  cosines  of  these  angles  are  the 
direction  cosines  of  OP. 

These  angles  are  interdependent,  and  the  result  of  this 
theorem  shows  that  the  sum  of  the  squares  of  their 
cosines  is  unity. 

Cor.  The  position  of  a  point,  P,  in  space  is  known 
relatively  to  the  origin  0,  and  the  axes  OX,  OY,  OZ, 
when  we  are  given  OP,  and  the  angles  which  OP  makes 


THE   OCTAHEDRON. 


87 


with  the  axes ;  or  when  we  are  given  the  length  of  the 
projections  of  OP  upon  the  axes.  For  the  projections 
are  the  direction  edges  of  a  cuboid  of  which  OP  is  the 
diagonal. 

This    is    the    fundamental    principle    entering    into 
analytic  spatial  geometry. 


The  Octahedron. 


99.  The  octahedron  may  assume  a  variety  of  forms, 
but  we  shall  confine  ourselves  to  those  in  which  the 
point  of  intersection  of  the  axis  is  the  middle  point  of 
each  axis,  or  the  centre  of  the  figure. 

In  general  the  octahedron  is  the  reciprocal  of  the 
parallelepiped,  formed  by  joining  the  centres  of  adjacent 
faces. 

The  three  joins  of  the  centres  of  opposite  faces  of  the 
ppd.  in  pairs  are  the  axes  of  the  octahedron,  and  hence 
from  the  nature  of  a  ppd.  (Art. 
37)  the  octahedron  may  be 
triclinic,  diclinic,  monoclinic, 
right  or  regular;  the  right 
octahedron  coming  from  the 
cuboid,  and  the  regular  from 
the  cube. 

100.  Theorem.  In  any  oc- 
tahedron, the  sum  of  the 
squares  on  the  twelve  edges 
is  equal  to  twice  the  sum  of 
the  squares  on  the  three  diameters  AA',  BB',  and  CC. 


88 


SOLID  OR   SPATIAL  GEOMETKY. 


Proof.     The  section  along  any  two  diameters,  being  a 
parallelogram,  gives 

AA''-  +  jBjB'2  =  AB"  +  BA^  +  A'B^  +  BA\ 

BB"  +  CC"  =  BC"  +  C'B"  -\-B'C'  +  CB% 

CC""  +  AA"  =  AC  +  CA"  +  A'C"  +  CA\ 

Whence,  by  addition, 

2  (^^'2  +  BB^  +  (7C"2)  =  2  2d'  =  2e2. 

Cor.     If  the  octahedron  is  regular,  all  the  edges  are 
equal  and  all  the  diameters  are  equal,  and  therefore 

and  the  section  ACA'C  is  a  square. 


The  Regular  Dodecahedron. 

101.   Let  AE,  AB,  and  AG  be  the  three  edges  which 
meet  to  form  the  corner  of  a  regular  dodecahedron.     Let 


Q  be  the  centre  of  the  face  ADB,  and  0  be  the  centre  of 
the  circumscribed  sphere. 


THE  REGULAR   DODECAHEDRON.  89 

Since  all  the  faces  are  congruent,  BEG  is  an  equi- 
lateral triangle,  and  OA  passes  through  its  centroid  P,  and 
is  normal  to  the  plane  of  the  triangle  (Art.  79.  Cor,  1). 

Z  ABE=  36°,  and  BE  =2BH=2AB  cos  36°=2e  cos  36°. 

Then,  BP=^BE  -  ^3  =  ^e^3  cos  36°. 

Also,    •.  •  BPA  =  1 ,  AP'  =  AB'  -  BP\ 

or  ^p2=e-(l-|cos2  36°). 

But  if  AA  be  a  diameter  of  the  circumsphere,  ABA^ 
is  a  "1,  since  B  is  on  the  sphel-e. 

.-.  AA'-AP=AB''  (Art.  P.  169°)  ; 

or  2R-AP=e\ 


Whence,        R  = 


ieV3 


Vl3-4cos2  36°| 
e^3 


Or  R= — — ^  =fi  v1.4012.^R... 

4  V  sin  6° .  sin  66° 

Again,  we  have. 


.-.  /3  =  eVl  1-401258  -0.25} 
=  6X1.309016..., 
and  r  =  ^{R^-Ag'l. 

But  AQ  =  ^ABsecBAQ 


—  1^'      „„   CAO  —  I 


cos  54°      '   sin  36° 


.-.  r=ej\  1.401258'  -  —r\^,  ] 
\  1  4sin2  36°  J 


=  6X1.113516 


90  SOLID   OR   SPATIAL   GEOMETRY. 


EXERCISES   F. 

1.  Two  opposite  edges  of  a  tetrahedron  are  perpendicular  to 
one  another  when  of  the  remaining  edges  the  sums  of  the  squares 
upon  opposite  edges,  taken  in  pairs,  are  equal. 

2.  What  does  the  theorem  of  Art.  91  become  when  the  four 
vertices  of  the  tetrahedron  become  complanar  ? 

3.  What  does  the  theorem  of  Art.  92  become  when  D  comes  to 
the  centroid  of  the  triangle  ABC? 

4.  Show  that  the  tangent  of  the  angle  made  by  an  edge  of  a 
regular  tetrahedron  with  one  of  the  faces  is  -^2. 

5.  In  the  cube,  P  is  the  middle  point  of  AB,  and  S  is  the 
middle  point  of  A'B' ;  show  that  the  acute  angle  of  the  section 
through  Q,  D,  Sis  cos-UvlO- 

6.  In  the  cube,  DK  is  ±  from  D  upon  the  diagonal  BB' ;  show 
that  DK  =  I  e  \/6  ;  and  that  CK  =  e. 

7.  In  the  cube,  the  join  of  the  middle  point  of  AB  with  B', 
and  the  join  of  the  middle  point  of  AD  with  D',  divide  each  other 
into  parts  which  are  as  2  : 1. 

8.  The  angle  between  two  diagonals  of  a  cube  is  cos"i  ^. 

9.  In  the  cube,  the  angle  between  a  diagonal  and  a  face  is 
1   1 

10.  In  the  cuboid,  the  angle  subtended  at  the  centre  by  the 
middle  points  of  two  conterminous  edges  is 


cos-i  a2  /  V(a2  +  62) (a2  +  c^), 
with  variations  in  the  letters  for  the  different  cases. 
11.    In  the  cuboid,  the  angle  between  diagonals  is 
cos-i  (a^  -  &^  -  c2)  /  (a2  +  62  +  c2) , 
with  symmetrical  variations. 


THE  KEGULAR  DODECAHEDRON.  91 

12.   In  the  cuboid,  the  ±  from  a  vertex  upon  a  diagonal  is 


a  Vfe^  +  c2  /  Va"^  +  62  +  c2, 
with  symmetrical  variations. 

13.  In  an  octahedron,  there  may  be,  at  most,  six  different 
lengths  of  edges. 

14.  If  the  semi-diameters  of  an  octahedron  be  a,  b,  c,  and 

Z  (be)  =\,   Z  (ca)  =  n,  and  Z  (ab)  =  v, 
then  the  squares  of  the  edges  are 

a^  +  b'^±2ab  cos  v,  b^  +  c^  ±  2  6ccos \,   c^  +  a^  ±2  ca cos /x. 

15.  In  a  right  octahedron,  the  cosines  of  the  dihedral  angles  are 
62c2  +  c2a2  -  a^^,  c'^a'^  +  a%^  -  b^c'^,  a'^b'^  +  b^d^  -  c^a^, 

each  divided  by  a^b^  +  ftV  4.  c^a^, 

16.  In  a  regular  octahedron,  the  perpendicular  from  the  centre 
upon  a  face  is  I  eV6. 

17.  lu  a  regular  otcahedron,  the  cosine  of  a  dihedral  angle 
is  -  J. 

18.  The  section  through  the  middle  points  of  AC,  AB',  B'C, 
CA',  A'B,  and  B'C  is  a  hexagon  with  opposite  sides  parallel,  and 
is  regular  if  the  octahedron  is  regular. 

19.  A  section  of  an  octahedron  parallel  to  any  face  is  a  hexagon. 

20.  The  radius  of  the  tangent  sphere  to  the  edges  of  a  regular 
octahedron  is  |  e. 

21.  The  squares  of  the  radii  of  the  three  spheres  of  a  regular 
octahedron  are  in  harmonic  proportion. 

22.  In  a  regular  dodecahedron, 

iil=^(Vl5  +  V3). 
4 


92  SOLID   OR   SPATIAL   GEOMETRY. 

23.  In  a  regular  dodecahedron, 

and  P=|(3  +  v5). 

24.  In  a  regular  dodecahedron,  if  D  be  the  dihedral  angle, 

sin  D=  1^5. 

25.  In  the  regular  dodecahedron  show  that  11 B^  exceeds  ISr" 
by  3e2. 

26.  In  the  icosahedron,  B  =  e-J(      »       ); 

27.  In  the  icosahedron  if  D  be  the  dihedral  angle, 

cos  D  =  I  y/5,  or  sin  Z>  =  f . 

28.  A  sphere  touches  one  face  of  a  regular  tetrahedron  exter- 
nally, and  the  three  others  internally.  Show  that  its  radius  is 
I j) ;  and   that  the  distance  from  the  further  vertex  at  which  it 

touches  the  three  faces  is  -  y/3. 

o 

29.  If  a  regular  cube  and  octahedron  be  circumscribed  to  the 
same  sphere,  their  vertices  are  conspheric. 

30.  If  a  regular  dodecahedron  and  icosahedron  be  circumscribed 
to  the  same,  sphere,  their  vertices  are  conspheric. 


SECTION  2. 

The  Sphere. 

102.  Def.  If  P  be  any  point,  and  a  line  through  P 
meets  a  given  sphere  in  A  and  B,  the  rectangle  PA  •  PB 
is  called  the  power  of  the  point  P  with  respect  to  the 
given  sphere. 

Cor.  A  point  is  without  a  sphere,  on  the  sphere,  or 
within  it,  according  as  the  power  of  the  point  with 
respect  to  the  sphere  is  positive,  zero,  or  negative. 

103.  The  power  of  a  fixed  point  with  respect  to  a 
given  sphere  is  independent  of  the  direction  of  the  line 
whose  segments  form  the  rectangle  which  measures  the 
power. 

Proof.  Let  the  line  through  P  meet  the  sphere  in 
A  and  B.  Since  P,  A,  B  are  in  line,  P,  A,  B,  0  are 
complanar,  0  being  the  centre  of  the  sphere. 

The  section  by  this  plane  is  a  great  circle,  with  a 
secant  line  through  P  cutting  the  circle  in  A  and 
B.  And  PA  '  PB  is  constant  in  value  for  this  circle 
(P.  Art.  176).  And  since  all  great  circles  have  the  same 
centre  and  equal  radii,  PA  •  PB  is  constant  for  every 
great  circle,  and  therefore  for  the  sphere. 

Cor.  If  A  and  B  become  coincident,  the  secant  line 
becomes  a  tangent,  and  the  rectangle  PA  •  PB  becomes 
the  square  on  the  tangent. 

93 


94 


SOLID   OR   SPATIAL  GEOMETRY. 


Therefore,  the  power  of  an  external  point  with  respect 
to  a  given  sphere  is  the  square  on  the  tangent  from  the 
point  to  the  sphere;  and  all  tangents  from  the  same 
point  to  the  same  sphere  are  equal. 


104.   S  and  S'  are  two  circles  with  centres  A  and 
B  and   radical  axis    L 
(P.  Art.  178). 

Let  the  whole  system 
revolve  about  the  com- 
mon centre-line  AB  as 
an  axis,  while  retaining 
the  fixed  relations  of  the 
several  parts. 

The  circles  describe 
spheres,  and  the  radical 
axis,  L,  describes  a  plane 
normal  to  AB. 

Also  PE-  PD  =  PE' '  PD'  remains  true  for  the  spheres. 
And  since  P  may  be  any  point  on  the  plane  described  by 
L,  the,  power  of  P  with  respect  to  each  sphere  is  the 
same. 

Def.  The  locus  of  a  point  of  Avhich  the  power  is  the 
same  with  respect  to  two  given  spheres  is  the  radical 
plane  of  the  spheres. 

Cor.  1.  Evidently,  the  radical  plane  of  two  spheres 
is  normal  to  the  join  of  their  centres,  and  divides  the 
distance  between  the  centres  so  that  the  diiference  of 
the  squares  on  the  two  parts  is  equal  to  the  difference 
of  the  squares  on  the  conterminous  radii. 


THE   SPHERE.  95 

Cor.  2.  The  tangents  to  two  spheres,  from  any  point 
on  their  radical  plane,  are  equal. 

Cor.  3.  The  plane  of  the  circle  of  intersection  of  two 
spheres  is  their  radical  plane. 

105.  Let  Si,  S2,  S^,  Si  be  four  spheres,  and  let  U12 
denote  the  radical  plane  of  >S'i  and  S2,  etc. 

The  four  spheres  have  the  six  radical  planes,  U12,  Uis, 
Uu,  U23,  11.24,  and  r/aj. 

A  point  whose  power  with  respect  to  Si  and  S2  is  the 
same  is  on  the  plane  U12,  and  a  point  whose  power  with 
respect  to  Si  and  S^  is  the  same  is  on  the  plane  Uis. 
Therefore,  a  point  whose  power  with  respect  to  Si,  S2, 
and  Ss  is  the  same  is  on  the  common  line  of  U12  and  Uis, 
and  is  evidently  on  the  plane  C/'ag. 

Therefore,  the  radical  planes  of  three  spheres  have  a 
common  line,  and  from  any  point  on  this  line  tangents 
to  the  spheres  are  equal. 

We  shall  call  this  line  the  radical  line  of  the  three 
spheres.  In  a  section  through  the  centres  of  the  spheres, 
this  line  gives  the  radical  centre  of  the  three  resulting 
great  circles. 

Cor.  1.  The  radical  line  of  three  spheres  is  normal  to 
the  plane  through  their  centres. 

Cor.  2.  The  six  radical  planes  to  four  spheres  inter- 
sect by  threes  to  form  four  axial  pencils. 

The  axes  of  these  pencils  may  be  denoted  by  L123,  Lm, 
ii34,  and  i234>  -^123  being  the  common  line  to  C/I2,  C^,  and 
U,i. 

Cor.  3.  The  line  ^123  meets  the  plane  Uu  in  one  point 
only,  and  it  evidently  meets  Uii  and  Uu  in  the  same  point. 


96  SOLID   OR   SPATIAL   GEOMETRY, 

Therefore,  there  is,  in  general,  one  point  from  which 
tangents  to  four  given  spheres  are  equal ;  or  of  which 
the  power  is  the  same  with  respect  to  four  given  spheres. 
This  is  the  radical  centre  of  the  four  spheres. 

EXERCISES   G. 

1.  Two  secants  are  drawn  through  the  same  point,  P,  within  a 
sphere,  and  meet  the  sphere  in  A,  B,  and  C,  D  respectively.  Then 
PA  ■PB=^PC-  PD. 

2.  If  a,  b  be  the  parts  into  which  the  plane  of  a  small  circle 
divides  the  diameter  through  its  centre,  the  area  of  the  small  circle 
is  wab. 

3.  If  three  spheres  intersect  two  and  two,  the  planes  of  the 
small  circles  of  intersection  form  an  axial  pencil. 

4.  If  four  spheres  intersect  two  and  two,  the  planes  of  the 
circles  of  intersection  pass  through  a  common  point. 

5.  Where  is  the  radical  centre  of  four  spheres  whose  centres 
are  complanar  ? 

6.  Under  what  condition  will  four  spheres  have  a  line  of  radical 
centres  ? 

(The  spheres  are  then  coaxal.) 

7.  The  tangent  cones,  common  to  three  spheres  taken  two  and 
two,  have  their  vertices  coUinear. 

8.  The  tangent  cones,  common  to  four  spheres  taken  two  and 
two,  have  their  vertices  complanar. 

9.  If  P  and  Q  be  two  points  iu  the  line  L,  and  U  and  V  inter- 
secting in  M  be  the  polar  planes  of  P  and  Q  with  respect  to  a 
sphere,  then  every  plane  through  M  is  polar  to  some  point  in  L ; 
and  L  and  M  are  perpendicular  to  each  other. 

10.  Any  rectilinear  figure  has  a  corresponding  rectilinear  figure 
such  that  every  side  in  the  first  figure  has  a  side  perpendicular  to  it 
in  the  second. 


Part  III. 

STEREOMETRY  AND  PLANIMETRY;  OR  THE  MEAS- 
UREMENT OF  VOLUMES  AND  SURFACES. 

106.  A  closed  spatial  figure  includes  within  its  boun- 
daries a  portion  of  space  separated  from  all  other  parts  of 
space.  This  portion  of  space  considered  with  respect  to 
extent,  and  not  with  respect  to  form,  is  called  the  volume 
of  the  closed  figure. 

As  our  primary  ideas  of  a  spatial  figure  were  probably 
derived  from  concrete  objects  such  as  blocks  of  wood  or 
stone,  the  volume  of  a  spatial  figure  is  also  called  its 
solid  contents,  and  the  figure  itself  is  called  a  solid.  Hence 
the  name  Solid  Geometry. 

Also  considering  a  closed  spatial  figure  as  a  surface, 
which  after  the  manner  of  a  closed  vessel  might  be  filled 
with  a  liquid,  the  volume  is  sometimes  called  the  capacity 
of  the  figure. 

The  measuring  of  volumes,  or  solid  contents,  or  capaci- 
ties is  called  Stereometry. 

Def.  Equal  spatial  figures  are  those  which  have  equal 
volumes,  and  therefore  congruent  figures,  when  having 
volumes,  are  necessarily  equal. 

97 


SECTION  1. 

POLYHEDRA. 

107.   Theorem.      Two   cuboids   with  congruent  bases 
have  their  volumes  proportional  to  their  altitudes. 

B'AGD  and  F-EGH 
are  two  cuboids  having 
their  bases  congruent.  Then 
vol.  B  .  ACD  :  vol.  F-  EGH 
=  BD.FH. 


D 


J 
1' 


Proof.      If   this   propor- 
tion is  not  true,  let 

vol.  B-AOD:  vol.  F ■  EGH  =  BD :  FI, 
where  FI  is  different  in  length  from  FH ;  and  first  let 
FI  be  less  than  FH.     As  a  general  case  let  BD  and  FH 
be  incommensurable. 

Take  some  u.l  (P.  Art.  150.  3)  less  than  IH  which 
will  measure  BD,  and  divide  BD  and  FH  into  parts 
equal  to  this  u.l.  One  point  of  division,  at  least,  must 
fall  at  some  point,  J,  between  /  and  H. 

Through  all  the  points  of  division  pass  planes  parallel 
to  the  bases.  These  divide  the  cuboids  B  •  ACD  and 
F  •  EG  J  into  congruent  and  therefore  equal  cuboids. 

.-.  vol.  B .  ACD :  vol.  F  •  EG  J  =BD:FJ 
and  vol.  B- ACD -.vol.  F' EGH  =BD.FI   (hyp.). 

.-.  vol.  F-EGJ:  vol.  FEGH=FJ:  FL 


POLYHEDRA.  99 

But         yo\.F-EGJ<Yo\.F-EGH; 

.-.  FJ  is  <FI; 

which  is  not  true. 

Hence  FI  cannot  be  less  than  FH.  And  in  like  manner 
it  is  shown  that  FI  cannot  be  greater  than  FH;  and 
as  FI  has  some  value,  it  must  be  equal  to  FH;  and 
therefore 

vol.  B  •  ACD  :  vol.  F  •  EGH=  BD  :  FH. 

Cor.  Cuboids  which  have  two  dimensions  in  each  re- 
spectively equal  have  their  volumes  proportional  to  their 
third  dimensions;  or,  more  generally,  a  cuboid  with  con- 
stant base  has  its  volume  varying  as  its  altitude. 

108.  Theorem.  Two  cuboids  are  to  one  another  as  the 
continued  product  of  their  three  dimensions. 

Let  X,  Y  denote  two  cuboids  whose  dimensions  are 
respectively  abc,  and  a'b'c'. 

Then  X:  T=  abc:  a'b'c'. 

Proof.  Let  Pbe  a  cuboid  whose  dimensions  are  a,  6,  c', 
and  Q  be  a  cuboid  whose  dimensions  are  a,  b',  c'. 

Then  X  and  P  have  the  face  ab  the  same,  and  P  and 
Q  have  the  face  ac'  the  same,  and  Q  and  T  have  the 
face  b'c'  the  same. 

.-.  X:P  =  c:c',  (Art.  107.  Cor.  1.) 

P:Q  =  b:b', 

Q  :  Y=  a:  a'. 

Whence,  by  compounding  the  three  proportions, 

X;Y==  abc -.a'b'c'. 


100  SOLID   OR   SPATIAL   GEOMETRY. 

Cor.  1.  A  generalised  statement  of  the  theorem  is, 
the  volume  of  a  cuboid  varies  as  the  continued  product 
of  its  three  dimensions. 

Cor.  2.    When  the  cuboids  are  similar,  their  homologous 

edges  are  proportional,  and  if  a',  b',  c'  be  homologous  to 

a,  b,  c, 

abc  _a?  _W  _c? 

a'b'c'~aP~V^~7^' 

Therefore,  two  similar  cuboids  are  to  one  another  as 
the  cubes  upon  two  homologous  line-segments. 

Or,  the  volume  of  a  cuboid  of  constant  form  varies  as 
that  of  the  cube  on  any  one  of  its  line-segments. 

109.  In  measuring  volumes  we  take  as  a  unit  the 
volume  of  the  cube  whose  edge  is  the  unit-length.  This 
volume  is  the  unit-volume,  and  it  will  be  denoted  by  u.v. 

The  three  units  of  extension  are  thus  interconnected, 
so  that  the  giving  of  any  one  of  them  gives  all. 

Thus  if  a  cube  has  its  edge  taken  as  the  u.l,  the  area 
of  one  of  its  faces  is  the  u.a.,  and  its  volume  is  the  u.v. 

If  the  edge  of  a  cube  be  n  unit-lengths,  each  of  its 
faces  contains  n^  unit-areas,  and  its  volume  contains  n^ 
unit-volumes  (comp.  P.  Art.  151). 

110.  Theorem.  The  number  of  u.v.s  in  a  cuboid  is 
the  continued  product  of  the  numbers  of  u.l.s  in  its  three 
direction  edges,  or  its  three  dimensions. 

Proof.  Let  X,  Y  be  the  cuboids  having  their  three 
direction  edges  expressed  by  a,  b,  c  and  a',  b',  c'. 

Then  X :  Y=  abc  :  a'b'c'.  (Art.  108.) 


THE   POLYHEDRA.  101 

Now  let  a'  =  6'  =  c'  =  one  u.l.  Then  Y  contains  one 
u.v.     And  hence : 

The  nnmber  of  u.v.s  in  X=the  number  of  u.l.s  in 
a  X  the  number  of  u.l.s  in  6  x  the  number  of  u.l.s  in  c. 

Tliis  result  is  generally  expressed  by  saying  that  the 
volume  of  a  cuboid  is  the  product  of  its  three  dimen- 
sions, an  expression  of  which  the  full  meaning  is  given 
above. 

Cor.  If  a,  b,  c  be  the  three  direction  edges  of  a  cu- 
boid, ab  denotes  the  area  of  the  face  whose  edges  are 
a  and  b,  and  c  is  the  altitude  to  that  face  taken  as 
base. 

Therefore,  the  volume  of  a  cuboid  is  the  product  of 
the  area  of  its  base  multiplied  by  its  altitude. 

111.  The  product  form  of  three  quantitative  symbols, 
where  the  symbols  denote  line-segments,  is  to  be  inter- 
preted as  the  volume  of  the  cuboid  having  for  its  three 
direction  edges  the  line-segments  denoted  by  these 
symbols. 

Hence  such  expressions  as  abc,  (a-\-b)ab,  etc.,  denote 
volumes  of  cuboids  in  geometry,  and  are  consequently 
said  to  be  of  three  dimensions  even  in  algebra. 

This  exhausts  the  geometry  of  space  as  we  know  it, 
for  space  has,  for  us  at  least,  only  three  dimensions. 

112.  Expressions  such  as  abed,  or  a'b^,  or  a^bc,  etc., 
are,  in  algebra,  said  to  be  of  four  dimensions ;  but  when 
the  letters  are  line-symbols,  no  interpretation  is  possible 
in  real  geometry. 

They  may  be  then  said  to  belong  to  a  hypothetical 
or  imaginary  something,  which  to  us  can  have  no  real 


102  SOLID   Oil   SPATIAL   GEOMETRY. 

existence,  but  which  is  spoken  of  as  geometry  of  four 
dimensions,  or  as  space  of  four  dimensions. 

In  using  the  symbols  and  forms  of  algebra  to  deduce 
geometric  relations,  expressions  of  four  or  of  higher 
dimensions  may  occur  as  intermediate  steps  in  some 
transformation,  but  never  as  iinal  results. 

The  associative  law  in  algebra  which  tells  us  that 
a  •  be,  ab  '  c,  b  •  ac  are  equal,  tells  us  in  geometry  that  the 
measure  of  the  volume  of  a  cuboid  is  independent  of 
which  face  is  taken  as  a  base. 

The  expression  a^b  is  the  cuboid  having  the  square  on 
side  a  as  base  and  b  as  altitude,  or  the  cuboid  having  the 
rectangle  ab  as  base  and  a  as  altitude ;  and  these  are  the 
same  cuboid  differently  viewed. 

The  forms  ■\/abc,  -y/a^b,  etc.,  are  not  geometrically 
interpretable ;  but  -yjabcd  is  an  area. 

The  form  ^abc  denotes  a  line-segment,  the  edge  of 
the  cube  whose  volume  is  equal  to  the  cuboid  whose 
dimensions  are  a,  b,  c. 

Parallelepiped. 

113.  Theorem.  A  parallelepiped  is  equal  to  the  cuboid 
which  has  its  base  and  altitude  respectively  equal  to 
those  of  the  parallelepiped. 

We  prove  this  theorem  by  showing  that  any  parallele- 
piped can  be  transformed,  without  change  of  volume, 
into  a  cuboid  having  a  base  and  altitude  equal  to  those 
of  the  ppd.     Let  A  •  A'BD  be  a  triclinic  ppd. 

Cut  it  by  a  plane,  EFO,  normal  to  the  direction  edge  AA'. 

This  section  is  a  parallelogram,  and  BFE,  CGF,  FEA', 
etc.,  are  Is. 


PARALLELEPIPED. 


103 


r//7  f// 

^/fj  /I 

/ ''  v 

/  H 

In 

F  _,' 

i/  ^°' 

\/ 

On  AA}  produc.ed  take  A'E'=AE,  and  through  E'  pass 
the  plane  E'F'G'  II  to  EFG. 

Then  the  corners  A-BDE  and  A' ■  B'D'E'  are  evi- 
dently congruent,  since  they  are  composed  of  equal  face 
angles  disposed  in  the  same 
order.  And  if  the  figure 
A  •  BGE  be  so  placed  that 
A  coincides  with  A',  AD 
with  A'D',  and  AB  Avith 
A'B',  this  figure  will  co- 
incide completely  with 
A'-B'G'E',  and  the  ppd. 
AC  is  transformed  to  the 

monoclinic  ppd.  EG',  without  change  of  volume,  and 
the  base  EH'  is  equal  to  the  base  AD',  and  the  altitude 
remains  unchanged. 

Again,  by  passing  a  plane  normal  to  the  direction  edge 
EH  of  the  monoclinic  ppd.  we  transform  it  into  a  cuboid 
in  which  the  volume  is  unchanged,  and  the  base  and 
altitude  are  unchanged. 

Therefore  any  ppd.  can  be  transformed,  without  change 
of  volume,  into  a  cuboid  having  its  base  and  altitude 
equal  to  those  of  the  ppd. 

Therefore  a  ppd.  is  equal  to  the  cuboid  having  its 
base  and  altitude  equal  to  those  of  the  ppd. 

Note  .  —  If  the  ppd.  is  such  that  it  is  impossible  to  cut  it  by  the 
plane  EFG,  normal  to  AA',  then  EFG  may  be  any  plane  less 
inclined  to  AA'  than  the  face  ABC  is.  We  thus  transform  the 
ppd.  into  another  triclinic  ppd.  less  oblique  than  the  original ;  a 
second  section  may  now  be  made  normal  to  a  direction  edge ;  or 
if  not  a  second,  a  third,  etc. 


104 


SOLID   OR   SPATIAL   GEOMETRY. 


Cor.  1.  The  volume  of  a  parallelepiped  is  the  product 
of  the  area  of  its  base  by  its  altitude. 

Cor.  2.  Similar  ppds.  are  to  each  other  as  the  cubes 
on  homologous  line-segments. 

Cor.  3.  A  ppd.  of  constant  form  varies  as  the  cube  on 
any  of  its  line-segments. 

Prism. 

114.  If  a  cuboid  or  a  monoclinic  ppd.  be  divided  into 
two  triangular  prisms  by  a  plane  passing  through  a  pair 
of  opposite  edges,  which  are  normal  to  a  face,  the 
prisms  so  formed  are  congruent,  and  therefore  equal. 
But  if  a  plane  be  passed  through  opposite  edges  of  a  tri- 
clinic  ppd.,  the  two  prisms  formed  are,  in  general,  not 
congruent,  but  symmetrical,  and  they  cannot  therefore 
be  shown  to  be  equal  by  superposition.  We  proceed  to 
show  that  they  are,  however,  equal. 

115.  Theorem.  The  two  triangular  prisms  into  which 
a  parallelepiped  is  divided  by  a  plane  through  a  pair  of 
opposite  edges,  are  equal. 

Let  A .  BDA'  be  a  tri- 
clinic  ppd.  A,  A',  C,  C 
are  complanar,  and  their 
plane  divides  the  ppd. 
into  the  triangular  prisms 
A'BCA'&nd  C-ADC. 

These  prisms  are  not 
congruent.    But,  as  in  Art. 

113,  transforming  the  triclinic  ppd.  into  the  monoclinic 
ppd.  E  •  FHE',  we  transform,  without  change  of  volume. 


PRISM.  105 

the  prism  A-BCA'  into  E-FGE',  and  C  -  ADC  into 
G  •  EHO'.  And  these  new  prisms,  being  right  prisms 
from  the  monoclinic  ppd.,  are  congruent,  and  therefore 
equal,  and  each  is  one-half  the  monoclinic  ppd. 

Therefore,  the  original  prisms  A  •  BCA'  and  E  •  FGE' 
are  equal,  and  each  is  one-half  the  triclinic  ppd. 

Cor.  1.  Since  the  right  section  EFGH  is  double  the 
right  section  EGH,  it  follows  that  the  volume  of  a  prism 
is  the  area  of  a  right  section  multiplied  by  the  length 
of  a  lateral  edge. 

Cor.  2.  Taking  ABCD  as  the  base  of  the  ppd.,  and 
ACD  as  the  base  of  the  prism  C  •  ADC,  these  figures 
have  the  same  altitude. 

Therefore  (Art.  113.  Cor.  1),  the  volume  of  a  prism  is 
the  area  of  the  base  multiplied  by  the  altitude. 

Cor.  3.  As  all  prisms  may  be  divided  into  triangular 
prisms,  Cors.  1  and  2  are  true  for  all  prisms. 

116.  We  have  two  expressions  for  the  volume  of  a 
prism : 

1st,  vol.  =  area  of  rt.  section  x  lateral  edge. 

2d,  vol.  =  area  of  the  base     x  altitude. 

area  of  rt.  section  _     altitude 
area  of  base  lateral  edge 

But  if  AA'  be  the  lateral  edge,  and  AP  be  the  alti- 
tude, AP  ^  AA'  is  the  cosine  of  the  angle  between  the 
lateral  edge  and  the  altitude. 

But,  as  the  altitude  is  normal  to  the  base,  and  the 
lateral  edge   is   normal  to  a  right   section,  this  is  the 


106  SOLID   OR   SPATIAL   GEOMETRY. 

angle  between  a  right  section  and  the  plane  of  the  base. 
Calling  this  the  angle  of  obliquity  of  the  prism,  we  have : 
The  area  of  a  right  section  of  a  prism  is  equal  to  the 
area  of  an  oblique  section  multiplied  by  the  cosine  of 
the  angle  of  obliquity. 

Of  Laminje. 

117.  To  fix  our  ideas,  let  AETN  be  a  trapezoid  with 
^r  II  to  AN.  Divide  its  side  AE  into  any  number  of 
equal  parts,  and  through  the  points  of  division,  B,  C,  D, 
etc.,  draw  lines  II  to  AN. 

On  these  lines  and  the  bases 
-4^and  ET construct  the  series 
of  internal  rectangles  BP,  CQ, 
DR,  ES,  •••  and  the  series  of 
external  rectangles  Aa,  Bb,  Cc, 
Dd-'. 

The   area   of    the    trapezoid 
evidently  lies  between  the  sum  of  the  external  rectan- 
gles and  the  sum  of  the  internal  rectangles. 

Now,  any  external  rectangle  as  Cc  is  congruent  with 
an  internal  rectangle  below  it,  CQ ;  except  that  the 
lowest  external  rectangle  has  no  corresponding  and  con- 
gruent internal  one,  and  the  uppermost  internal  rectan- 
gle has  no  congruent  external  one. 

Let  E  denote  the  sum  of  the  external  rectangles,  and 
I  denote  the  sum  of  the  internal  ones.     Then 

E-I=nAa-nES. 

.'.  the  difference  between  the  sum  of  the  external 
rectangles  and  the  sum  of  the  internal  rectangles  is  less 


E 

T 

d 

D 

/ 

I-    -rC 

h 

c 

/ 

S 

\ 

B 

/ 

R 

\ 

t  — tC 

y 

Q 

M. 

OF   LAMINJE.  107 

than  the  lowermost  external  rectangle ;  and  this  is  true 
however  many  rectangles  be  formed. 

But  the  lowermost  rectangle  can  be  made  as  small  as 
we  please,  by  making  its  altitude  sufficiently  small ;  i.e. 
by  making  the  number  of  parts  into  which  we  divide 
AE  sufficiently  great.  And  hence  the  area  of  the  trap- 
ezoid is  the  limit  of  the  sum  of  either  series  of  rectan- 
gles as  the  number  of  rectangles  is  indefinitely  increased. 

118.  Now,  let  AETN  be  a  vertical  section  of  a  frus- 
tum of  a  pyramid  (Art.  59),  in  which  AN  and  ET  are 
sections  of  the  bases.  Divide  AE  into  any  number  of 
equal  parts,  and  through  the  points  of  division  pass 
planes  parallel  to  the  bases. 

On  the  figures  of  section  construct  a  series  of  inscribed 
prisms,  BP,  CQ,  DR,  ES  •••,  and  a  series  of  circum- 
scribed prisms,  Aa,  Bb,  Cc,  Dd'-. 

The  volume  of  the  frustum  lies  between  the  sum  of 
the  internal  prisms  and  the  sum  of  the  external  prisms. 

But  any  external  prism,  except  the  lowermost,  has  a 
congruent  internal  prism  below  it,  and  any  internal 
prism,  except  the  uppermost,  has  a  congruent  external 
prism  above  it. 

Hence  if  E  denotes  the  sum  of  the  external  prisms, 
and  /  of  the  internal  prisms, 

E  —  1=  prism  Aa  —  prism  ES 

=  vol.  of  lowermost  external  prism 

—  vol.  of  the  uppermost  internal  prism. 

And  this   is  true,  however  many  equal  parts  AE  is 
divided  into. 


108  SOLID   OR   SPATIAL   GEOMETRY. 

Therefore,  the  volume  of  the  frustum  differs  from  the 
sum  of  either  series  of  prisms,  by  less  than  the  volumes 
of  the  series  of  prisms  differ  from  each  other  ;  that  is, 
by  a  quantity  less  than  the  lowermost  external  prism ; 
and  this  difference  may  be  made  as  small  as  we  please 
by  dividing  AE  into  a  sufficiently  large  number  of  parts. 

Hence,  the  volume  of  the  frustum  is  the  limit  of  that 
of  either  series  of  prisms,  when  the  number  of  prisms 
is  indefinitely  increased. 

Coi:  This  theorem  is  exceedingly  important,  for  the 
least  consideration  will  show  that  nothing  in  the  inves- 
tigation requires  that  AE,  or  any  edge,  should  be  a 
straight  line,  and  hence  that  the  theorem  holds  true 
when  the  boundary  of  the  figure,  between  the  parallel 
bases,  is  composed  partly  or  wholly  of  curved  surfaces ; 
also  that  the  theorem  is  true  when  one  or  both  bases 
reduce  to  lines  or  points. 

119.  Def.  When  a  spatial  figure  is  cut  by  two  indef- 
initely near  parallel  planes,  the  prism,  having  one  of 
the  sections  as  base,  and  the  distance  between  the  planes 
as  altitude,  is  called  a  lamina  of  the  spatial  figure. 

When  two  figures  are  confined  between  the  same  two 
parallel  plants,  the  laminae  determined  by  two  indefinitely 
near  planes,  parallel  to  the  confining  planes,  are  corre- 
sponding laminoe. 

Usually  the  planes  which  determine  a  lamina  are 
supposed  to  be  infinitely  near,  so  that  a  lamina  is  one  of 
the  prisms  of  the  preceding  article,  taken  at  its  limit. 

Cor.  1.  From  Art.  118,  it  appears  that  two  figures 
which  have  all  corresponding  laminae  equal  are  them- 


THE  PYRAMID. 


109 


selves  equal;  and  two  figures  whicli  have  all  corre- 
sponding laminae  in  the  same  ratio  are  themselves  in 
that  ratio,  the  one  to  the  other. 

Cor.  2.  Since  corresponding  laminae  have  the  same 
altitude,  their  volumes  are  proportional  to  their  bases  ; 
and  hence  corresponding  laminae  are  equal  when  corre- 
sponding sections  are  equal ;  and  corresponding  laminae 
are  in  the  same  proportion  to  one  another  as  are  the  cor- 
responding sections. 

The  Pyramid. 

120.  Theorem.  Pyramids  are  equal  whose  bases  are 
equal  and  whose  altitudes  are  equal. 

Proof.  Let  the  trian- 
gular pyramids,  D  •  ABG 
and  H'EFG,  have  their 
bases  equal,  and  also  their 
altitudes  equal,  and  let 
them  be  so  placed  that 
their  bases  are  compla- 
nar,  and  their  vertices  are 
upon  the  same  side  of  this 
plane.  Then  D  and  H  lie 
in  a  plane  parallel  to  the 
plane  of  the  bases. 

Let  abc  and  efg  be  corresponding  sections. 

Then  (Art.  28.  Cor.  2) 

Aabc^AABC,a.ndAefg^AEFG. 
But  (P.  Art.  218.  2),      A  abc:  A  ABC  =  aW :  AB?. 


110  SOLID   OR   SPATIAL   GEOMETRY. 

And  since  DA  and  DB  are  cut  by  parallel  planes,  AB 
is  II  ab. 

.'.  ah^ :  AB""  =  Da?  :  DA\ 

And         (Art.  27)  Da:  DA  =  He:  HE; 

.-.  al/  :  AB"  =  Da^ :  DA'  =  He- :  HE'  =  ef :  EF% 

or  Aabc:AABC=Aefg:AEFG. 

But  AABC=AEFG;  (hyp. ) 

.  •.  A  abc  =  A  efg. 

And  as  corresponding  laminae  are  equal,  tlie  volumes 
of  the  pyramids  are  equal  (Art.  119.  Cor.  1). 

And  since  all  pyramids  may  be  divided  into  triangu- 
lar pyramids, 

Therefore,  any  two  pyramids  are  equal  whose  bases  are 
equal  and  whose  altitudes  are  equal. 

Cor.  Two  frustums  of  pyramids  which  have  their 
two  bases  respectively  equal  and  their  altitudes  equal 
are  themselves  equal. 

121.  Theorem.  A  triangular  prism  can  be  divided  into 
three  equal  pyramids. 

Proof.  A  -  BCD  is  a  triangular  prism.  Pass  a  plane 
through  the  points  A,  C,  and  E,  and  another  plane  through 
G,  D,  and  E. 

The  prism  is  divided  into  three  equal  pyramids. 

For  C  •  FDE  and  E  •  CAB  have  their  bases  DEF  and 
ABC  equal,  and  their  altitudes  the  same  as  that  of  the 
prism.     These  pyramids  are  therefore  equal  (Art.  120). 

Also  the  pyramids  C  •  ADE  and  C  •  ABE  have  their 


THE   PYRAMID. 


Ill 


bases  ADE  and  ABE  equal  (P.  Art.  141.  Cor.  1),  and 

have  their  vertices  coincident. 
Therefore  they  have  the  same 
altitude  and  are  equal. 

And  the  prism  is  thus  divided 
into  three  equal  pyramids. 


Cor.  1.  As  each  triangular  pyr- 
amid is  one-third  of  the  corre- 
sponding triangular  prism,  and  as 
every  prism  can  be  divided  into 
triangular  prisms ; 

Therefore  every  pyramid  is  one-third  of  the  prism 
having  the  same  base  and  altitude  as  the  pyramid. 

Cor.  2.  If  jB  denotes  the  area  of  the  base  of  a  pyramid, 
and  h  denotes  its  altitude, 

vol.  of  pyramid  =  ^  KB. 

Cor.  3.  Pyramids  with  equal  bases  are  to  one  another 
as  their  altitudes,  and  pyramids  with  equal  altitudes  are 
as  their  bases. 

122.  Theorem.  The  frustum  of  a  triangular  pyramid 
may  be  divided  into  three 
pyramids,  two  of  which  have 
the  bases  of  the  frustum  as 
their  bases,  and  the  altitude 
of  the  frustum  as  their  alti- 
tude, and  the  third  of  which 
is  a  mean  proportional  between 
the  first  two. 

ABCDEF  is   a   triangular 
frustum. 


0^ F 


112  SOLID   OR   SPATIAL   GEOMETRY. 

The  plane  through  A,  E,  F  cuts  off  the  pyramid 
A  •  DEF,  whose  base  DEF  is  the  upper  base  of  the 
frustum. 

From  the  remaining  figure  the  plane  through  A,  E,  C 
cuts  off  the  pyramid  E  •  ABC,  whose  base  ABC  is  the 
lower  base  of  the  frustum. 

We  have  left  the  pyramid  E  •  AFC. 

Join  BF  and  CD. 

The  pyramids  B  •  AEC  said  F-  AEC  having  the  com- 
mon base  AEC  are  as  their  altitudes,  and  the  altitudes 
are  as  PB  to  PF,  or  BC  to  EF. 

.:  BAEC:F.AEC=BC:EF. 

Again,  the  pyramids  C-AEF  (which  is  the  same  as 
F-AEC)  and  D-AEF  having  the  common  base  AEF 
are  to  one  another  as  CQ  is  to  QD,  or  AC  to  DF. 

But,  since  the  bases  are  similar  (Art.  28.  Cor.  2), 
BC :  EF  =  AC :  DF. 

.-.  B .  AEC :  F-AEC  =  F-  AEC :  D •  AEF. 

Or  the  pyramid  F-  AEC,  or  C-  AEF,  is  a  mean  propor- 
tional between  the  pyramids  E  -  ABC  and  A  •  DEF. 

Cor.  If  B  and  B'  denote  the  bases  of  the  frustum, 
and  h  the  altitude, 

vol.  of  E-ABC=\nB,  vol.  of  A-DEF=\hB', 
and    . •.  vol.  oiF-AEC=^h  VBB'. 

The  volume  of  the  frustum  is  accordingly  : 

vol.  =  ^hlB  +  B'-\-ViB>]. 

123.  The  volume  of  the  frustum  may  also  be  found  as 
follows : 

Let  0  be  the  vertex  of  the  pyramid  from  which  the 


THE  PYRAAHD. 


113 


frustum  is  formed,  and  let  OP  be  the  altitude  of  the 
pyramid.  Also  let  OP'  be  the  altitude  of  the  pyramid 
0  •  DEF,  Avhich  is  removed  in  forming  the  frustum.    Then, 

The  frustum  =  pyr.  O-^BC- pyr.  O-DEF, 

and  OP-OF  =  li. 

Since   any  area   may  be  expressed   as   a   square,  let 
b-=BoT  the  base  ABC,  and  b'^  =  B'  or  the  base  DEF. 

Then  OP:  OP' =  OA:  0D  =  AB:  DE  =b:b', 

and      OP-OP':OP=b-b':  b. 


OP{b  -  b')  =  bh. 


and 


...  0P=-^,  and  0P'  = 
b-b'' 

frust.  =^OP'b--^OP' 


b'h 
b-b' 
b" 


=  i/t 


(^iz:y)=Wb'-  +  b''+bb') 

=  \h{B  +  B'+VBB'). 


Cor.  The  volume  of  a  frustum  of  a  pyramid  is  the 
sum  of  the  two  bases  and  a  mean  proportional  between 
the  bases,  multiplied  by  one-third 
of  the  altitude. 

124.  Def.  A  triangular  prism 
with  non-parallel  bases  is  called  a 
truncated  triangular  prism,  or  a 
wedge. 

Let  ABC,  DEF  be  the  bases  of 
the  wedge,  of  which  ABC  is  nor- 
mal to  the  lateral  edges. 

Through  D  pass  a  plane  II  to 
ABC  and  draw  AP  ±  to  BC. 


114 


SOLID   OPw   SPATIAL   GEOMETRY. 


The   wedge   =   the   prism   A  •  BCD  +   the   pyramid 
DEFGH.     But  the  prism  =  A  ABC  x  AD;   and  the 

^^^^^"^  =  i^P  X  trapezoid  EH, 

=  I A  ABC{BE+  CF-2  AD)  ; 

.-.  vol.  of  wedge  =  \  A  ABC  {AD  +  BE  +  CF). 

Or  if  e„  62,  63  denote  the  edges,  and  B  the  area  of  a 
right  section, 

vol.  =1^(61  +  62  +  63). 

125.   Theorem.     If  a  tetrahedron  be  cut  by  a  plane 
which  bisects  two  edges  and  passes  through  an  opposite 
vertex,  the  volume  of  the  tetra- 
hedron is  equal  to  four-thirds  of 
the  prism  having  the  section  as 
base,  and  the  perpendicular  from 
any  other  vertex  on  the  plane  of 
section  as  altitude. 

A  •  BCD  is  a  tetrahedron,  and 
E  and  F  are  middle  points  of 
AB  and  AC  respectively. 

AP  is  perpendicular  upon  the 
plane  EFD. 

Then  \EF  is  II  to  BC,  and  bisects  AB,  EF  is  one-half 
BC,  and  AAEF=IAABC  (P.  Art.  218.  2). 

The  pyramids  having  these  triangles  as  bases  have  D 
as  a  common  vertex; 

.-.  tetrahedron  A  •  BCD  =  4  tetr.  A  ■  DEF 


=  ^AP-ADEF. 


Q.  E.  D. 


prismatoid  and  allied  foems.  115 

Prismatoid  and  Allied  Forms. 

126.  Def.  A  polyhedron  with  two  parallel  polygonal 
bases,  and  all  its  lateral  faces  plane  rectilinear  figures, 
and  all  its' lateral  edges  the  joins  of  vertices  of  opposite 
bases,  is  2i  prismatoid. 

This  definition  includes  the  prism,  pyramid,  and  frus- 
tum of  a  pyramid  as  special  cases,  and  is  more  general 
than  any  of  these. 

When  none  of  the  faces  are  triangles,  the  figure  is  the 
frustum  of  a  pyramid,  or  a  prismoid,  according  as  the 
lateral  edges  are,  or  are  not,  concurrent  when  produced. 

127.  ABCD  and  EFG  are  parallel  bases  of  a  prisma- 
toid, and  AEB,  EBF,  FBC,  CFG,  etc.,  are  triangular 
faces,  which,  in  the  figure  given,  are  seven  in  number. 

G 


A 

If  n  denotes  the  number  of  sides  in  one  base,  and  n' 
in  the  other,  it  is  readily  seen  that  the  number  of  faces 
cannot  be  greater  than  n  +  n'. 


116  SOLID   OR   SPATIAL  GEOMETRY. 

But  if  an  edge  of  one  base  be  connected  by  lateral 
edges  with  a  parallel  edge  of  the  other  base,  two  trian- 
gular faces  become  a  quadrangular  face,  and  the  whole 
number  of  faces  is  reduced  by  one.  Thus,  if  EF  were 
parallel  to  AB,  the  edges  AE,  EB,  and  BF  would  be 
complanar,  and  the  two  triangular  faces  AEB  and  BEF 
would  become  one  quadrangular  face,  AEFB. 

If  two  other  edges  of  the  bases  become  parallel,  a  like 
reduction  may  take  place,  and  the  whole  number  of  faces 
be  reduced  by  two. 

And  finally,  if  the  bases  have  the  same  number  of 
sides,  and  each  edge  in  one  base  be  connected  with  a 
parallel  edge  in  the  other,  all  the  faces  become  quadran- 
gular, and  the  figure  becomes  a  frustum  of  a  pyramid  or 
a  prismoid,  according  as  the  edges,  when  produced,  are 
or  are  not  concurrent. 

Even  with  the  same  bases,  however,  the  general  appear- 
ance of  the  figure  will  vary  with  the  different  ways  of 
connecting  the  vertices  of  the  bases  by  the  lateral  edges. 

128.  Def.  Take  H,  I,  J,  etc.,  middle  points  of  the 
lateral  edges,  AE,  BE,  BF,  etc.,  respectively. 

Since  HI  is  parallel  to  AB  (P.  Art.  84.  Cor.  2.  2)  and 
IJ  is  parallel  to  EF,  and  JK  to  BO,  etc.,  it  follows  that 
H,  I,  J,  etc.,  lie  in  a  plane  which  bisects  all  the  lateral 
edges,  and  is  parallel  to  the  bases.  The  section  by  this 
plane  is  called  the  middle  section. 

The  middle  section  contains,  at  most,  n  -f-  n'  sides, 
there  being  always  as  many  sides  as  there  are  faces  in 
the  prismatoid. 

The  middle  section  may  contain  re-entrant  angles, 
although  no  such  angles  are  found  in  either  base;  and 


PRISMATOID  AND  ALLIED  FORMS.  117 

it  will  frequently  have  such  angles  when  the  bases  are 
polygons  of  different  species,  or  when  their  vertices  are 
connected  in  some  particular  order. 

Cor.     The  middle  section  bisects  the  altitude. 

129.  Volume  of  the  prismatoid.  Take  P,  any  point  in 
the  plane  of  the  middle  section,  and  join  it  to  A,  D, 
E,  H,  and  N.    Denote  the  altitude  of  the  prismatoid  by  h. 

G 


A 
Then,  P  -  ADE  is  a  tetrahedron,  and  PNH  is  a  section 
through  a  vertex,  P,  and  the  middle  points,  H  and  N, 
of  two  opposite  edges. 

.-.  vol.  of  P '  ADE  =  I  /i  X  A  PNH.     (Art.  125.) 
Similarly,  by  joining  P  to  all  the  remaining  vertices, 
B,  C,  F,  etc.,  and  to  the  remaining  middle  points,  7,  J,  K, 
etc.,  we  have, 

Sum  of  all  the  tetrahedra  of  which  P-ADE  is  the 
type  =  I A  X  (the  sum  of  the  A  of  which  PNH  is  the 
type),  or 

^(P'ADE)  =  ya(APNH). 


118  SOLID   OR   SPATIAL  GEOMETRY. 

But  S  (A  PNH)  =  the  area  of  the  middle  section,  and 
denoting  the  area  of  the  middle  section  by  M, 

'2,{P-ADE)  =  ^hM. 

Now,  after  removing  all  these  tetrahedra,  we  have  left 
two  pyramids  having  P  as  a  common  vertex,  and  the 
bases  of  the  prismatoid  as  their  respective  bases.  The 
altitude  of  these  prisms  being  ^h  (Art.  128.  Cor.),  their 
volumes  are  ^hB  and  ^hB',  where  B  and  B'  are  the 
areas  of  the  bases  of  the  prismatoid. 

.-.  vol.  of  prismatoid  =  - (B-\- B' +  4:M). 
6 

Cor.  The  prismatoid  is  equal  to  four  pyramids,  two 
having  the  bases  of  the  prismatoid  as  their  bases  and  half 
the  altitude  of  the  prismatoid  as  their  altitude,  and  two 
having  the  middle  section  as  their  bases  and  the  altitude 
of  the  prismatoid  as  their  altitude. 

Cor.  The  formula  of  the  present  article  is  known  as 
the  prismoidal  formula.  On  account  of  its  extremely 
wide  range  of  applicability  it  is  the  most  important  of 
all  formulae  connected  with  the  determination  of  the 
volumes  of  the  more  prominent  spatial  figures. 

The  following  examples  are  some  illustrations  of  its 
application. 

(a)  Prism.  Here  the  two  bases  and  the  middle  sec- 
tion are  all  congruent. 

Hence,  vol.  =  ^  {B+B  +  4  jB)  =  hB.    (Art.  115.  Cor.  2.) 

(&)  Pyramid.  The  upper  base  vanishes,  and  the  mid- 
dle section  is  one-fourth  the  lower  base. 


EEGULAR  POLYHEDRA.  119 

.-.  vol.  =  ^  (5  +  0  +  5)  =  I  /iS.         (Art.  121.  Cor.  2.) 

(c)  Frustum  of  a  pyramid. 

Let  B,  B'  be  the  bases,  and  M  be  the  middle  section. 
And  since  any  area  may  be  expressed  as  a  square,  let 
B  =  h%  B'  =  b",  and  M=  m\ 

Then         2m=b  +  b'.       ■. 

.'.  4:m^  =  4:M=b'-\-b'^  +  2bb' 

=  B-\-B'  +  2VBB'. 

.:  \(B  +  B'  +  4.M)=-{2B  +  2B'  +  2VBB') 
6  6 

=  I  (B  +  -B'  +  -\/BB').     (Art.  122.  Cor.) 

(d)  Tetrahedron,  in  terms  of  a  middle  section  (Art. 
61.  Def.  1)  and  the  length  of  the  common  perpendicular 
to  the  edges  parallel  to  the  section. 

In  this  case,  which  has  an  important  subsequent  appli- 
cation, both  bases  vanish,  and  we  have 

vol.  =  I  liM. 


Regular  Polyhedra. 

130.  A  regular  polyhedron  of  n  faces  is  divisible  into 
n  congruent  pyramids  whose  bases  are  the  several  faces 
of  the  polyhedron,  and  whose  altitude  is  the  radius  of 
the  in-sphere  to  the  polyhedron. 


120  SOLID  OR   SPATIAL   GEOMETRY. 

Hence,  if  n  be  the  number  of  faces,  B  be  the  area  of 
a  face,  and  r  be  the  radius  of  the  in-sphere,  we  have 

vol.  =  -Br. 
3 

(a)    Regular  Tetrahedron. 

B  =  — -^3,  r  =  ^e-y/6,  and7i  =  4. 

•••vo1.=|-|V3-tV«V6 

Cor.  As  the  expression  for  the  volume  may  be  writ- 
ten  -  ( 1 ,  therefore  the  cube  on  the  side  of  a  square 

whose  diagonal  is  the  edge  of  a  regular  tetrahedron  is 
three  times  the  tetrahedron. 

(6)    Regular  Octahedron. 

5  =  ^eV«%  r  =  \e^Q,  «  =  8. 

.•.vol.  =  f.i.i.eV18  =  ieV2. 

Cor.     This  volume  may  be  written  ^(e-^2)'. 

Therefore,  the  cube  on  the  diagonal  of  a  square  whose 
side  is  the  edge  of  a  regular  octahedron  is  six  times  the 
octahedron. 

(c)    Regular  Dodecahedron. 

By  the  methods  of  plane  geometry,  we  find  the  area 
of  a  regular  pentagon  with  side  e  to  be 


-i^'V(l^7 


BEGITLAR  POLYHEDRA.  121 

=  |(15  +  7V5). 
(d)    Eegular  Icosaliedroii. 

3     4\^       8       J 
=  Ae3(3+V5). 


EXERCISES  H. 

1.  If  a  plane  parallel  to  the  bases,  and  midway  between  them, 
be  passed  through  the  prism  of  Art.  121,  compare  the  areas  of  the 
sections  of  the  three  pyramids. 

2.  Apply  the  conditions  of  Ex.  1  to  Art.  122. 

3.  A  plane  of  section  passes  through  the  middle  points  of  the 
parallel  edges  of  a  wedge,  one  of  whose  bases  is  a  right  section 
(Art.  124).     Find  the  area  of  the  section. 

4.  If  Ci,  e.i,  63  be  the  three  parallel  edges  of  a  wedge,  show  that 
1(^1  +  «2  +  63)  is  the  distance  between  the  centroids  of  the  bases. 

5.  Apply  the  prismoidal  formula  to  find  the  volume  of  a  wedge. 

6.  A  prismoid  has  both  bases  parallelograms  with  angle  6,  and 
the  sides  are  a,  b  for  the  one,  and  a',  b'  for  the  other.  Find  its 
volume,  its  altitude  being  A. 

7.  Show  that  the  cube  on  the  side  of  a  square  whose  diagonal  is 
the  edge  of  a  regular  octahedron  is  three-fourths  of  the  octahedron. 


122 


SOLID  OR   SPATIAL   GEOMETRY. 


8.  If  a  regular  tetrahedron  and  a  regular  octahedron  have  the 
same  edge,  the  octahedron  is  four  times  the  tetrahedron. 

9.  AA',  BB',  CC,  DD',  being  diagonals  of  a  cube,  show  that 
the  plane  through  DBC  cu^s  off  a  pyramid  whose  volume  is  one- 
sixth  that  of  the  cube. 

10.  The  direction  edges  of  a  cuboid  are  a,  ft,  c,  and  a  plane 
passes  through  the  three  distal  extremities  of  these.  Show  that 
the  area  of  the  section  is  jVa^fez  _|_  j^IqI  _^  (^if^z^ 

11.  AA\  BE',  etc.,  are  the  diagonals  of  a  ppd.  Show  that  a 
plane  through  DBC  cuts  off  a  pyramid  which  is  one-sixth  the  ppd. 

12.  The  direction  edges  of  a  ppd.  are  a,  ft,  c,  and  the  angles 
between  them  are  Z.  (be)  =  \,  Z  (ca)  =  n,  Z  (aft)  =  v.  Then  the 
vol.  is 

abc  y'  { 1  —  cos^  X  —  cos^  fi  —  cos^ «/  +  2  cos  X  cos  fj.  cos  v). 

OA,  OB,  OC  are  the  direction  edges; 

Z  COB  =  X,  Z  COA  =  At,  ZAOB  =  v. 

Let  CP  be  normal  to  the  plane  of  A  OB, 
and  Pg,  PB  be  ±s  upon  OA  and  OB. 

vol.  of  ppd.  =  OA-  OB  sin  v  •  CP. 

(P.  Art.  215.)    o^ 

OQPB  are  coney clic,  and  OP  is  a  diameter 
of  the  circumcircle ; 


OP 


_  QB 


(P.  Art.  228.) 


and 

But 
and 


CP  =  y/l  OC'^  -  ^^]  =  -^  V(c2  sin2  V  -  QB^)  ; 
\  sm2  vj     sm  V 

vol.  =  ab  VCc^  sin2  v  -  QB^). 

Qm  =  0^2  +  0P2  -20q-  OBcosv; 

0§2  =  c2  cos2  fi,  and  0P2  -  c^  cos2  X. 


REGULAR   POLYHEDRA.  123 

Whence,  by  substitution, 

vol.  =  abc  v'{  1  —  cos2  X  —  cos2 /x  —  cos^  i/  +  2 cos X  cos  /u  cos  v}} 

13.  Show  from  the  character  of  the  result  in  Ex.  12  that  if  in 
j.ny  ppd.  X,  n,  V  are  all  acute,  all  vertices  except  the  one  opposite 
0  have  one  acute  and  two  obtuse  angles,  etc. 

14.  With  the  vertices  of  a  ppd.  as  centres,  equal  spheres  are 
described  to  cut  the  ppd.  Then  the  volume  removed  by  all  the 
spheres  is  equal  to  that  of  one  of  the  spheres. 

15.  Show  that  space  may  be  wholly  divided  up  into  regular 
octahedrons  and  tetrahedrons,  and  that  there  will  be  twice  as  many 
of  the  latter  as  of  the  former. 

1  The  area  of  a  parallelogram  whose  sides  are  a,  h  and  angle 
0  \s  ah  s,m.e,  or  ab-yj{l  —  cos^^),  and  the  volume  of  the  ppd.  is 
dbCy/(\  —  cos"^x  —  cos^ju  —  etc.).  On  account  of  the  analogy  in 
form,  the  expression  1  —  cos^x  —  cos^^u  —  etc.  is  sometimes  called 
the  square  of  the  sine  of  the  solid  angle  O  •  ABC,  and  it  usually 
appears  in  the  matrix  form 

1  cos  X      cos  IX 

cos  X .      1        cos  V 
cos  IX    cos  V        1 

The  analogy,  however,  is  one  of  form  only,  as  there  are  no  func- 
tions of  solid  angles  really  corresponding  to  the  sine,  cosine,  tan- 
gent, etc.,  of  plane  angles. 


SECTION   2. 

Cone,  Cylinder,  Sphere. 

The  Cone. 

131.  The  cone  of  Art.  67  is  not  a  closed  figure,  and 
consequently  does  not  admit  of  measurement  for  volume. 
But  if  the  cone  be  cut  by  a  plane  which  does  not  pass 
through  the  centre,  and  which  makes,  with  the  axis,  an 
angle  greater  than  the  vertical  angle,  a  closed  figure  is 
formed  by  the  conical  surface  and  the  plane.  It  is  this 
closed  figure  -that  is  called  a  cone  in  relation  to  stere- 
ometry. 

The  centre  of  the  cone  is,  in  this  relation,  called  the 
apex  or  vertex,  and  that  portion  of  the  section  plane 
which  forms  a  part  of  the  enclosing  figure  is  the  base  of 
the  cone. 

The  word  'cone,'  whenever  having  reference  to  stereo- 
metrical  relations,  will  mean  this  figure. 

132.  As  the  director  curve  may  be  of  any  form,  and 
as  the  plane  of  section  may  assume  different  relative 
directions,  the  variations  in  the  cone  are  unlimited. 

If  the  cone  be  circular,  and  the  plane  of  section  be 
perpendicular  to  the  axis,  the  figure  is  the  right  circular 
cone ;  and  this  is  the  most  important  of  all  the  cones. 

The  base  is  a  circle,  and  the  axis  of  the  cone  passes 
through  the  centre  of  the  circle. 

124 


THE   CYLINDER.  125 

A  right  circular  cone  is  generated  by  a  right-angled 
triangle  while  revolving  about  one  of  the  sides  as  an  axis. 
The  other  side  then  generates  the  base  (Art.  9.  Cor.  1), 
and  the  hypothenuse  generates  the  convex  surface. 

133.  The  cone  may  be  looked  upon  as  the  limiting  form 
of  a  right  regular  pyramid,  when  the  number  of  sides  in 
the  base  is  indefinitely  increased,  and  the  length  of  each 
side  is  correspondingly  diminished. 

But  the  volume  of  any  pyramid  is  one-third  of  its 
altitude  multiplied  by  the  area  of  its  base ; 

Therefore,  the  volume  of  a  cone  is  one-third  of  its 
altitude  multiplied  by  the  area  of  its  base. 

Cor.  If  the  base  be  circular  and  its  radius  be  r,  its 
area  is  iri^.     And  if  h  be  the  altitude  of  the  cone,  the 

vol.   =  \  TTT^h. 

134.  The  frustum  of  a  cone  is  the  limit  of  the  frus- 
tum of  a  pyramid,  and  its  volume  is  therefore 

^/i(JS  +  -B'-|- V55'). 

But  if  r  and  r'  be  the  radii  of  the  bases, 

B  =  TTT^,  B'  =  7rr'2,  and  ■VbW=  nrr'. 

.-.  vol.  =  i7r/i  (7-2 +  r'2  +  rr'). 

The  Cylinder. 

135.  When  the  cylinder  of  Art.  75  is  cut  by  two  parallel 
planes  which  cut  completely  through  the  surface,  a  closed 
figure  is  formed,  which  is  the  cylinder  of  stereometry. 


126 


SOLID   OR   SPATIAL   GEOMETRY. 


When  the  planes  are  perpendicular  to  the  axis  of  the 
cylinder,  the  figure  is  a  right  cylinder.  Otherwise  it  is 
an  oblique  cylinder. 

136.  It  is  obvious,  from  the  definitions,  that  the 
cylinder  is  the  limiting  form  of  the  prism,  Avhen  the 
number  of  sides  in  the  base  is  indefinitely  increased  and 
the  lengths  of  each  side  correspondingly  diminished. 

Hence  the  measure  of  a  cylinder  is  the  area  of  the 
base  multiplied  by  the  altitude  (Art.  115.  Cor.  2). 

Cor.  If  the  cylinder  be  circular  and  right,  and  r  be 
the  diameter  of  the  base, 

vol.  =  irl^h, 
where  h  is  the  altitude. 

The  Sphere. 

137.  ABCD  is  a  tetrahedron  in  which  the  edge  AB 
is  equal  and  perpendicular  to  the  edge  CD,  and  KJ, 


joining  the  middle  points  of  these  edges,  is  the  common 
perpendicular  to  them. 


THE   SPHERE.  127 

Also,  K'BQT  is  a  sphere  having  its  diameter 

K'J'  =  KJ. 

We  shall  prove  that  corresponding  laminae  of  the  tet- 
rahedron and  of  the  sphere  are  in  a  constant  ratio,  by 
proving  that  corresponding  sections  are  in  a  constant 
ratio. 

Proof.  Let  parallel  planes  pass  through  AB  and  CD. 
Then  KJ  is  a  common  normal  to  these  planes,  "and  if  the 
sphere  be  placed  between  the  planes  with  K'J'  parallel 
to  KJ,  the  planes  will  touch  the  sphere  at  K'  and  at  J'. 

Let  the  sphere  and  tetrahedron  be  relatively  so  placed, 
and  let  EG  and  RQT  be  corresponding  sections  of  the 
figures  (Art.  119). 

Then  EFGHis  a  rectangle,  and  QRT  is. a  circle,  and 

KP  =  K'P. 

^j  KP     AE     EH 

^°"  KJ'=AD=DC' 

,  PJ'    DF     EF 

^^^  kj=^db  =  ab' 

.'.  by  multiplication 

KP  ■  PJ^  EF '  EH^  OEO 
KJ'  AB'  AB"  ' 

Also,  denoting  the  radius  of  the  sphere  by  r, 

K'P  '  PJ' ^  P'R^  ^  1  TT-P'R^^l  OQRT 
K'J"  K'J"     tt'     KJ'        it'     4/^    * 

Therefore     :•  KPPJ=  K'P '  -  P'J', 

CJEG      Am  .     . 

— = =  a  constant. 

OQRT     A-n-r' 


128  SOLID  OR   SPATIAL   GEOMETRY. 

Hence  the  corresponding  sections  of  the  tetrahedron 
and  of  the  sphere  are  in  a  constant  ratio ;  and  the  vol- 
umes of  the  tetrahedron  and  the  sphere  are  in  the  same 
ratio. 

Cor.  1.     The  tetr,  :  the  sphere  =  A^  :  47rrl 
But  the  tetr.  =  |  KJ  x  mid.  sec.      (Art.  130.  d.) 

=  \r-AB'^; 

.:  vol.  of  sphere      =|^7rr'. 

Cor.  2.  The  expression  for  the  volume  of  a  sphere 
may  be  written  f  •  2r  •  ttt^. 

But  2  r  is  a  diameter  of  the  sphere,  and  ttt^  is  the  area 
of  a  great  circle.  Therefore,  2r  •  ttt^  is  the  volume  of  the 
right  circular  cylinder  which  circumscribes  the  sphere. 

Hence  a  sphere  is  two-thirds  of  its  right  circumscribing 
cylinder. 

138.  As  the  prismoidal  formula  applies  to  any  portion 
of  the  tetrahedron  confined  between  planes,  each  parallel 
to  AB  and  CD,  and  since  laminae  of  the  sphere  hold  a 
constant  relation  to  corresponding  laminae  of  the  tetra- 
hedron, it  follows  that  the  prismoidal  formula  applies  to 
any  portion  of  the  sphere  limited  between  parallel  planes. 

Thus,  applying  the  formula  to  the  whole  sphere,  we 
have 

B  =  0,  B'  =  0,  M=  nr",  and  h  =  2r. 

.-.  vol.  =  —  (0 -f  0 -f  47rr2)  =  |7rr8. 


THE   SPHERE. 


129 


A  E 

O 


139.  Bef.  A  portion  of  a  sphere  enclosed  between 
two  parallel  planes  is  usually  called  a  zone  of  the  sphere ; 
but  if  one  of  the  planes  is  a 
tangent  plane,  the  zone  be- 
comes a  segment  of  the  sphere. 

140.  Volume  of  a  zone. 
Let  a  sphere  be  cut  by  par- 
allel planes,  given  in  section, 
in  the  diagram,  by  AB  and 
CD;  and  let  XY  denote  in 
section  the  plane  which  is  parallel  to  the  cutting  planes, 
and  half  way  between  them. 

The  data  usually  furnished  from  which  to  find  the 
volume  of  the  zone,  are  the  radii  CD  and  AB  of  the  two 
bases,  and  the  length  of  their  common  normal  AC,  or  the 
altitude  of  the  zone.  Hence  we  suppose  AB,  AC,  and 
CD  to  be  the  known  quantities. 

We  have,  0  being  the  centre  of  the  sphere, 

OA^  -f  AB'  =  OC  +  CD'  =  OX'  +  XT', 

since  each  expression  is  the  square  on  the  radius  of  the 
sphere. 

.-.  OA'  +  AB'  =  (0A  +  2 AXy  +  CD' 

=  {OA  +  AX)^  +  XT'. 

AB'  =  4.  AX'  +  CD' +  ADA' AX 
=  AX'  +  XY'  -\-2  0A'  AX. 
4  OA .  AX=AB^  -  CD'  -  4.  AX' 


Hence 


and  hence 


=  2AB^-2XY'-2AX'', 
2XY'  =  AB'  -\-CD'  +  2  AX'. 


130  SOLID   OR   SPATIAL  GEOMETRY. 

Now,  TT-  XY^  is  the  area  of  the  middle  section, 

and        TT  •  A^^  and  tt  •  CD^  are  the  areas  of  the  bases, 

and        2  AX  is  the  altitude  of  the  zone ; 

.-.  Yol.  =  ^ TT  -  AC  \3AB'  + 3  CD' +  AC'\. 

Or,  denoting  the  radii  of  the  bases  by  r  and  r',  and  the 
altitude  by  h, 

vol.=  i7rA[3?-2  +  3r'2  +  /i2|. 

Cor.  If  r  =  0,  the  zone  becomes  a  segment,  and  its 
volume  is  ^  7r^  (3  r^  +  h^) . 

141.    The  expression  for  the  volume  of  the  zone  may 
be  transformed  as  follows: 
Draw  DE  ±  to  AB. 

SABT'  +  3  (7Z>2  +  AC'=  2{AB  +  CD'-  +  AB  •  CD) 

+  {AB-CDY-\-AC^; 

and  \Tr'AC{AB'+CD''  +  AB.CD) 

is  the  volume  of  the  frustum  of  the  cone  which  has  the 
same  bases  and  altitude  as  the  zone. 

And  AC  being  the  projection  of  BD  on  OC,  if  we 
denote  the  angle  between  BD  and  AC  by  /8, 

•       AC  =BD  cos  13. 

.'.  |7r .  AC-BD""  =  J-TT  •  BD'cosjB 

=  sphere  on  BD  as  diameter  x  cos  /3. 

Therefore,  the  zone  exceeds  the  inscribed  conical  frus- 
tum by  the  sphere  on  the  slant  height  as  diameter 
multiplied  by  the  cosine  of  the  semi-vertical  angle  of 
the  cone. 


THE   SPHERE.  131 


EXERCISES   I. 

1.  Compare  the  volume  of  a  sphere  (1)  with  that  of  the  circum- 
scribed cube  ;  (2)  with  that  of  the  inscribed  cube. 

2.  Compare  the  volume  of  the  sphere  with  that  of  the  circum- 
scribed regular  tetrahedron. 

3.  A  cone  circumscribes  a  sphere  and  has  its  slant  height  equal 
to  the  diameter  of  its  base.  Show  that  vol.  of  cone  :  vol.  of  sphere 
=  9:4. 

4.  If  in  Ex.  3  a  plane  passes  through  the  circle  of  contact,  the 
vol.  of  cone  removed  is  f  the  vol.  of  sphere  removed. 

5.  A  cylinder  of  radius  a  passes  centrically  through  a  sphere 
of  radius  r.     Show  that  the  volume  removed  from  the  sphere  is 

I  irr^  (1  —  cos3  ^),  where  sin  ^  =  - . 

6.  A  circular  cone  with  semi-vertical  angle  a  has  its  vertex  at 
the  centre  of  a  sphere  of  radius  r.  Show  that  the  volume  common 
to  the  cone  and  sphere  is  |  in-^  (1  —  cos  a). 

7.  A  right  circular  cone  has  its  vertex  lengthened  out  into  a 
linear  edge  equal  and  parallel  to  a  diameter  of  the  base.  Show 
that  the  volume  is  one-half  that  of  the  circumscribing  cylinder. 
(The  resulting  figure  is  known  as  the  common  conoid.) 

8.  A  cone  whose  semi-vertical  angle  is  45°  has  the  diameter  of 
a  sphere  as  its  axis,  and  its  vertex  on  the  sphere.  Show  that  one- 
fourth  of  the  sphere  lies  without  the  cone. 

9.  The  cone  of  Ex.  8  has  its  semi-vertical  angle  equal  to  a ; 
then  the  part  of  the  sphere  lying  without  the  cone  is 

|irr3(l-f  cos2a)2. 


SECTION   3. 

142.  In  this  section  we  propose,  under  three  heads,  A, 
B,  and  C,  to  explain  and  illustrate  some  special  methods 
of  measuring  volumes,  by  applying  these  methods  to  the 
cone,  cylinder,  sphere,  and  some  other  spatial  figures. 

A.    Spatial  Figures  generated  by  the  Motion 
OF  A  Plane  Figure. 

143.  When  a  variable  plane  figure  moves  so  that  a 
fixed  point  lying  in  its  plane  describes  a  line  or  curve 
not  complanar  with  it,  the  plane  figure  describes  or 
generates  a  spatial  figure. 

The  plane  figure  is  then  the  generator,  and  the  line 
or  curve  is  the  path  of  the  particular  point  which  de- 
scribes it. 

The  case  as  here  stated  is  too  general  for  use,  especially 
in  elementary  geometry  or  by  elementary  methods.  We 
therefore  subject  the  elements  of  the  description  to 
certain  conditions,  usually  as  follows. 

(1)  The  generator  is  a  closed  plane  curve,  being  in- 
variable in  form,  while  being  either  variable  or  constant 
in  dimensions. 

(2)  The  path  is  a  line  normal  to  the  plane  of  the 
generator.     This  line  will  be  called  the  axis. 

(3)  The  generator  preserves  its  orientation,  i.e.  any 
fixed  line  of  the  generator  is  invariable  in  direction ;  or 

132 


GENERATED   FIGURES. 


133 


any  fixed  point  in  the  generator  describes  a  line  or  curve 
complanar  with  the  axis.  This  line  or  curve,  whose 
form  depends  upon  the  nature  of  the  variation  of  the 
generator,  is  a  guide  to  the  motion  of  the  generator,  and 
forms  the  director. 

Thus  if  the  nature  of  the  variation  of  the  generator 
is  given,  the  director  is  also  given ;  and  if  the  director 
is  given,  the  nature  of  the  variation  is  given. 

144.  Let  PQR  be  a  variable  circle,  whose  centre,  C, 
moves  along  the  fixed  line  AB  normal  to  the  plane  of 
the  circle.     AB  is  the  axis. 


(1)  Let  P,  any  point  on  the  circle,  be  guided  by  the 
fixed  director  line  L,  which  meets  AB  in  D. 

Then,  evidently  the  generating  circle  describes  a  cone 
having  D  as  vertex  and  AB  as  axis. 

The  radius  CP  is  in  a  constant  ratio  to  DC. 

Hence  a  variable  circle,  whose  centre  moves  on  a  fixed 
line  normal  to  its  plane,  and  whose  radius  varies  as  the 
distance  of  the  centre  from  a  fixed  point  in  the  line, 
describes  a  cone. 


134  SOLID   OR   SPATIAL   GEOMETRY. 

(2)  If  the  generating  figure  in  case  (1)  were  a  polygon, 
the  figure  generated  would  be  a  right  pyramid. 

(3)  Let  P  move  on  the  line  M  parallel  to  AB. 

The  circle  describes  a  cylinder,  and  a  polygonal  genera- 
tor describes  a  prism. 

(4)  Let  P  be  guided  by  the  circle,  Z,  tO  which  AB 
is  a  centre  line,  and  EF  a  diameter. 

The  circle  PQR  then  generates  a  sphere  whose  diam- 
eter is  EF. 

If  0  be  the  centre  of  the  director  circle,  it  is  evident 
that  CP2  +  CO"  =  OP''  =  constant. 

Therefore,  a  variable  circle,  whose  centre  moves  on 
a  line  normal  to  its  plane,  and  whose  radius  so  varies 
that  the  sum  of  the  squares  on  the  radius  and  on  the 
distance  of  the  centre  of  the  circle  from  a  fixed  point  in 
the  line  is  constant,  generates  a  sphere. 

(5)  If  the  generator  in  case  (4)  were  a  polygon,  the 
figure  generated  would  be  a  polygonal  groin;  the  most 
common  groin  is  the  square  one. 

In  a  similar  manner  many  other  figures  may  be  gener- 
ated, such  as  the  oblate  spheroid,  the  prolate  spheroid, 
the  hyperboloid,  the  paraboloid,  the  ellipsoid,  etc. 

145.  Consider  a  number  of  equidistant  points  along 
the  axis.  Let  the  generator  at  these  points  be  taken  as 
bases  of  prisms  or  cylinders  whose  altitudes  are  the  dis- 
tances between  consecutive  points. 

We  have  then  a  series  of  prisms  or  cylinders,  of  equal 
altitude,  inscribed  in  or  circumscribed  about  the  spatial 
figure,  as  the  case  may  be. 


GENERATED  FIGURES. 


135 


But  (Art.  118)  the  volume  of  the  spatial  figure  is  the 
limit  of  either  series  of  prisms  or  cylinders,  when  their 
number  is  indefinitely  increased  and  their  altitudes  cor- 
respondingly diminished. 

Hence  if  we  can  obtain  an  expression  for  the  total 
volume  of  any  number  of  such  elementary  prisms  or 
cylinders,  we  can  deduce  the  expression  for  the  volume 
of  the  spatial  figure,  by  imposing  the  condition  that 
the  number  of  elementary  prisms  or  cylinders  shall  be 
infinite. 

In  carrying  out  this  operation  we  assume  the  two  fol- 
lowing relations,  which  are  proved  in  almost  any  work 
on  algebra : 

{A)         '1+2+3  +...+W  =iw2+i7i; 

(B)  12  +  22  +  32+...+w2  =  ^n'  +  ^w2+in, 

where  n  denotes   any  positive   integer,  and   the   series 
extends  from  1  to  n. 


146.  Let  X  be  a  closed  plane  figure,  which  remains 
invariable  in  form  while  varying  its  dimensions. 

Let  a  given  point  P  be 
guided  by  the  line  AH,  and 
let  a  point  Q  move  on  AC. 

Then  X  describes  a  spatial 
figure,  a  cone  or  pyramid, 
having  some  position  of  the 
generator  at  B,  as  above. 

Let  X  denote  the  area  of 
the  variable  figure,  X,  at  any  stage  in  its  variation,  and 
let  B  denote  the  area  of  CDE,  the  final  stage  of  X. 


136  SOLID   OR   SPATIAL   GEOMETRY. 

Then  X.B  =  PQ" :  HC.        (P.  Art.  218.  6.) 

And  from  similar  triangles,  APQ  and  AHC, 
PQ':HC'=AP^:AH\ 

Denote  AH  by  h,  and  let  AH  be  divided  into  n  equal 
parts,  and  let  AP  be  m  of  these  parts. 

Then  AP  =  ~AH, 

n 

and  .-.  X=—z'B. 

But  the  elemental  cylinder,  or  prism,  on  X  as  base  has 

-  •  AH,  or  -,  as  its  altitude,  and  therefore  its  volume  is 
n  n 

Bh.% 
rr 

This  expresses  the  volume  of  any  element,  a  particular 
one  being  got  by  giving  a  particular  value  to  m.  m  =  1 
gives  the  first  element,  lying  next  A;  m  =  2  gives  the 
second,  etc.,  and  m  =  n  gives  the  last,  lying  next  H 

The  sum  of  these  elements  is 

3^|l'  +  2'  +  3'  +  -+»') 
(  Us  3 

This  holds  true  for  all  integral  values  of  n.  When 
we  go  to  the  limit  by  making  n  infinite,  the  fractions 


GENERATED   FIGURES. 


137 


—  and  — -  become  zero,  and  the  sum  of  the  elements 
2  71         6  w^ 

becomes  the  volume  of  the  spatial  figure  (Art.  145). 
.  • .  volume  =  ^  Bli. 
As  B  may  have  any  closed  form  whatever,  this  ex- 
presses the  volume  of  any  species  of  cone  or  pyramid 
which  forms  a  closed  spatial  figure. 

147.  Let  the  generating  figure,  X,  of  constant  form, 
but  variable  in  dimensions,  be 
guided  by  the  axis  OA,  and  by 
the  circular  quadrant  CQA  as  a 
director,  0  being  the  centre  of 
the  quadrant. 

Let  CDE  be  the  generator  in 
the  position  in  which  0  lies  in 
its  plane,  and  let  S  denote  the 
area  of  CED,  and  X  denote  the 
area  of  the  generator  in  any 
position. 

Then,  since  PQ  is  ±  to  OA, 

PQ^=Oq^-OP'=OC^ 
But  X:S  =  PQ^.OC^; 


x= 


_PQ'  o_o    OP' 


s  =  s- 


oc  oc 

Now,  denote  OA  by  r  and  divide  it  into  n  equal  parts, 
and  let  OP  be  m  of  these  parts. 


Then 


m 


OP=-r,  and  00  =r. 


x=s-—-s. 


138  SOLID   OR   SPATIAL   GEOMETRY. 

The  elemental  cylinder  having  X  as  base  has  -  for 

n 
altitude,  and  its  volume  is  therefore 


rS 
The  sum  of  these  is 


^^  f  1  +  1  +  1+  •••n  terms  _  r  +  2^+  ••■n"  | 
\  n  v?  ) 

(        3     2n     ^mr ) 

at  the  limit,  when  w  becomes  infinite. 

And  the  volume  generated  while  moving  over  the 
whole  diameter  is 

vol.  =  I  rS. 

The  value  of  this  expression  for  volume  depends  upon 
the  value  of  S. 

1.  If  /S*  is  a  circle,  its  area  is  irr^,  and  the  figure  gener- 
ated is  the  sphere. 

.-.  vol.  of  a  sphere  =  ^  Trt"'. 

2.  If  /S  is  a  square,  and  the  middle  point  of  its  side  is 
at  C,  the  area  is  4r*,  and  the  figure  is  the  common  groin, 

and  its 

vol.  =  I  r^ ; 

since  the  groin  extends  only  from  0  to  A. 

3.  If  /S  is  a  regular  hexagon  with  a  vertex  at  C,  we 
have  a  hexagonal  groin,  and  its  volume  is  r^-y/3. 


SPATIAL  FIGURES.  139 

148.  By  varying  the  form  of  the  generator,  and  also 
of  the  director  curve,  a  great  variety  of  spatial  figures 
may  be  described. 

1.  With  a  circle  as  director  and  an  ellipse  as  generator, 
QR  being  the  major  axis,  we  get  the  oblate  spheroid;  and 
with  QR  as  minor  axis,  the  prolate  spheroid. 

2.  With  an  ellipse  as  director,  and  major  axis  as  axis, 
and  an  ellipse  as  generator,  we  get  the  ellipsoid;  with 
circle  as  generator  we  get  the  prolate  spheroid. 

3.  With  parabola  as  director,  and  a  circle  as  generator, 
we  get  the  paraboloid  of  revolution  ;  and  with  ellipse  as 
generator  we  get  the  elliptic  paraboloid. 

EXERCISES  J. 

The  axes  of  an  ellipse  being  a  and  b,  its  area  is  irab. 

1.  Show  that  the  volume  of  a  prolate  spheroid  is  irab^,  where 
a>b. 

2.  Show  that  the  volume  of  an  oblate  spheroid  is  va^b,  where 
a>b. 

3.  In  the  figure  of  Art.   147,  if  CQA  were  a  quadrant  of  an 

PQ2        n  p2 

ellipse,  and    0A  =  a  and  OC  =  b,  then   =-^  +  -^=-=—  =  1.      Hence 

^  OC'2     OA^ 

find  the  volume  of  an  ellipsoid  when  the  axes  of  the  generating 
ellipse  are  b  and  c  at  the  position  S. 

4.  In  Art.  146,  if  PQ^  =  c  •  AP,  where  c  is  a  constant,  show  that 
the  volume  described  is  one-half  that  of  the  circumscribing  cylinder. 

5.  OC  is  an  axial  line  cut  by  a  curve  in  0  and  C,  and  PM  is  a 
perpendicular  from  a  point  P  on  the  curve  to  the  axis  OC  If 
PM  =  a{OM •  OC  —  OM'^),  show  that  the  volume  described  by  the 
curve  in  a  revolution  about  the  axis  is  ^^  of  that  of  the  circum- 
scribing cylinder  between  0  and  C 


140  SOLID   OR   SPATIAL   GEOMETRY. 


B.  Figures  of  Revolutioist. 

149.  When  a  plane  figure  revolves  about  an  axial  line 
lying  in  its  plane,  the  plane  figure  generates  a  spatial 
figure  bounded  wholly  or  partly  by  curved  surfaces, 
and  called,  from  its  mode  of  generation,  a  figure  of 
revolution. 

Under  the  same  circumstances  the  area  of  the  plane 
figure  generates  a  volume  of  revolution,  i.e.  the  volume 
of  the  figure  of  revolution. 

The  area  of  a  plane  figure  may  be  considered  as 
the  limit  of  the  sum  of  a  set  of  elements,  composed  of 
inscribed  rectangles  with  equal  but  indefinitely  small 
altitudes. 

In    revolution,   these    elements    of  area   describe   or 
generate  elements  of   volume,  whose   sum   has   for  its 
limit   the  volume   of   the   gen- 
erated spatial  figure. 


150.    Let  ^C  be  a  rectangle,         *^  " 
and   let   it   revolve   about   the         ^ 
axial  line  PR,  parallel  to  AD. 

The  volume  generated  by  the   

rectangle  AC  is  the  difference 

between  the  volumes  generated  by  PC  and  by  PD. 

But     the  vol.  by  PC  =  TT  •  PB'  •  BC, 

and         the  vol.  by  PZ)  =  tt  •  PA^  ■  BC; 

.'.  the  vol.  hy  AC  =7r-  BC(PB'  -  PA') 

=  TT  •  BC{PA  +  PB)  {PB  -  PA). 


FIGURES   OP   REVOLUTION".  141 

If  Q  be  the  middle  point  of  AB,  PQ  is  tlie  distance  of 
the  centre  of  the  rectangle  from  the  axis  of  revolution, 

^""^  PB-i-PA  =  2PQ; 

.'.  vol.  hj  AC=27r-PQ-BC-  AB, 

=  area  of  AC  x  the  circumference  of  the  circle  traced 
by  the  centre  of  AC. 

Therefore,  the  volume  described  by  a  rectangle  in  one 
revolution  about  an  axial  line  parallel  to  its  side,  and 
which  does  not  cross  the  rectangle,  is  the  area  of  the 
rectangle  multiplied  by  the  length  of  the  path  of  its 
centre. 

Cor.  1.  When  the  axial  line  passes  through  the  cen- 
tre of  the  rectangle,  the  length  of  path  described  by  that 
centre  is  zero,  and  hence  the  volume  described  is  zero. 

From  this  it  appears  that  if  a  revolving  plane  figure 
is  crossed  by  the  axis  of  revolution,  the  parts  of  the 
figure  lying  upon  opposite  sides  of  the  axis  generate 
volumes  which  must  be  taken  in  opposite  senses,  or  with 
opposite  signs. 

Cor.  2.  From  the  figure  we  have  2PQ  =  2PA  +  AB; 
and  hence  2Tr  •  PQ  =  27r  ■  PA +  Tr  •  AB. 

But  when  ^C  is  an  elemental  rectangle,  and  we  go  to 
the  limit  by  indefinitely  diminishing  AB,  PQ  has  for 
its  limit  either  PA  or  PB,  these  being  finally  the  same. 
Hence,  if  the  elemental  rectangle  AC  is  to  be  taken  at 
the  limit,  PA  may  be  taken  for  PQ. 

151.  Volume  of  a  cone  of  revolution.  The  rectangle 
AC  revolves  about  AB  as  an  axis. 


142 


SOLID   OR   SPATIAL  GEOMETRY. 


The  triangle  ACB  generates  a  cone  of  revolution ;  the 
rectangle  generates    a    cyl-    d 
inder;     and     the     triangle 
ACD  describes  that  part  of 
the  cylinder  which  remains 
after  the  cone  is  removed. 

On  retake  P,  Q,  any  near 
points,  which  at  the  limit 
become  coincident,  and  draw 
PB,  QS,  perpendicular  to  AB,  and  PT,  Q  V,  perpendicular 
to^Z>. 

Then  PS,  being  an  elemental  rectangle  of  the  triangle 
ACB,  generates  an  element  of  the  cone ;  and  PV,  in 
like  manner,  generates  an  element  of  the  portion  of  the 
cylinder  which  remains  after  removal  of  the  cone. 

But      vol.  of  element  hy  PS  =  ir  •  PR'  •  PE.    ■ 

And    vol.  of  element  by  PV=  7r{PB  +  FB)  PT  -  PF. 

And  from  similar  triangles  PBA  and  QEP, 


PB^QE  PB^PF 

BA     EP'     ^  PT     PE 

element  by  PS  _       PB 
element  by  PF 


PB  +  FB 

And  this  relation  being  true  for  any,  and  therefore  for 
every,  pair  of  corresponding  elements,  is  true  for  their 
sums. 

But  at  the  limit,  when  Q  comes  to  P,  PB  and  FB 
become  the  same. 

.-.  the  limit  of  ^(elements  by  PS)  ^  1. 
2 (elements  by  PF)      2 

Or,  cone  by  ABC :  figure  by  ACD  =  1:2. 


FIGURES   OF   KEVOLUTION. 


143 


Whence  it  follows  that  the  cone  is  one-third  of  the 
cylinder. 

Remark.  In  the  foregoing  investigation  we  might,  according 
to  Cor.  2  of  Art.  150,  have  taken  the  element  described  by  P Fas 
being  it  ■  2  PB  •  FT-  FF,  since  the  element  is  finally  to  be  taken  at 
its  limit. 

152.   Volume    of    a    sphere.      The    quadrant    DPA, 
and  its  circumscribed  square 
DBAC,  revolve  about  CA  as 
an  axis. 

The  quadrant  generates  a 
semisphere,  and  the  square 
generates  the  right  circum- 
scribed cylinder. 

On  the  arc  DA  take  P  and 
Q,  any  near  points  which  at 
the  limit  approach  to  coinci- 
dence. 

Draw  PR,  QS,  perpendicu- 
lars to  CD,  and  PT,  Q  V,  per- 
pendiculars to  DB.  Produce 
DC,  making  GG  =  DC. 

The  rectangle  PS,  being  an 
element  of  the  circle,  describes 
an  element  of  the  sphere,  and  the  rectangle  PV  for 
similar  reasons  describes  an  element  of  that  part  of  the 
cylinder  which  lies  without  the  sphere. 

The  volume  of  the  element  described  by  PS  is,  at  its 
limit  when  Q  comes  to  P,  2ir  •  CR-  PR-  RS ;  and  the 
volume  of  the  element  described  by  PF  is,  at  its  limit, 

-n^i.CR  +  CD)  '  PT ■  PF. 


144  SOLID  OR   SPATIAL  GEOMETRY. 

at  It   element  hj  PS ^2 PR  RS        CR 
'  element  by  PV      PT  '  PF'  CR  +  CD 

But  the  A  DPR  and  PGR  being  similar, 

PR^PR^GR^CR+CD 
PT     RD     PR         PR        ' 

And  the  A  PEQ  and  PRC  being  similar  at  the  limit 
when  Q  approaches  P, 

RS^PE^PE^PR 
PF     PF     EQ      CR 

,:  at  It.,  element  by  PS  =  2  x  element  by  PV. 

And  this  being  true  for  each,  and  therefore  every  pair 
of  corresponding  elements,  is  true  for  their  sums. 

Therefore  the  volume  generated  by  the  quadrant  is 
twice  the  volume  generated  by  the  figure  DPAB. 

Or  the  volume  of  a  sphere  is  two-thirds  that  of  the 
circumscribing  right  cylinder. 

Cor.  1.  If  r  be  the  radius  of  the  sphere,  the  volume  of 
the  circumscribing  cylinder  is  'n-7^-2r;  and  hence  the 
volume  of  the  sphere  is  ^ttt^. 

Cor.  2.  From  the  foregoing  investigation  it  follows 
that  wherever  Q  is  taken  on  the  arc,  with  CA  as  axis, 
the  volume  generated  by  the  segment  of  the  circle, 
DSQP,  is  two-thirds  the  volume  generated  by  the  rec- 
tangle D.SQF. 

153.  Volume  generated  by  an  isosceles  triangle  revolv- 
ing about  an  axis  which  passes  through  the  vertex  but 
does  not  cross  the  triangle. 


FIGURES   OF  REVOLUTION. 


145 


The  isosceles  A  OPQ,  with  PQ  as  base,  revolves  about 
the    axis    OD     passing 
through  the  vertex  0. 

LetPQmeetODinZ), 
and  draw  the  altitude 
OR,  and  project  P,  R,  Q, 
on  OD  at  A,  C,  and  B. 
Also  draw  QE  parallel 
to  OD. 

The  ZAPD=Z  ROD, 
and  hence, 

Therefore, 
also, 
and 

.-.  OD 

Now,  the  vol. 


APEQ^  AORD. 
OD'PE=OR-PQ', 
APEQ^AOCR, 
PQCR=OR-EQ', 
■  CRPE=  OR""  •  EQ  =  OR''  •  AB. 

described  by  A  OPQ  =  vol.  of  cone  by 
OPA  +  vol.  of  cone  by  DPA  —  vol.  of  cone  by  OQB 
—  vol.  of  cone  by  DQB, 

=  ^7r '  OD  {PA-  -  QBT')  =  \Tr-0D'2CR-PE 
=  f7r.  OR^'AB. 

Therefore,  the  volume  described,  in  one  revolution,  by 
an  isosceles  triangle  revolving  about  a  line  through  its 
vertex,  and  lying  without  it,  is  the  continued  product 
of  the  projection  of  the  base  of  the  triangle  upon  the  axis, 
the  area  of  the  square  on  the  altitude,  and  the  constant  |  tt. 

154.  Let  equidistant  points  A,  B,  C,  etc.,  be  taken  in 
the  arc  of  a  circle  of  which  0  is  the  centre  and  OL  is 
a  centre  line  not  crossing  the  arc. 

The  A  AOB,  BOO,  . . .,  are  all  isosceles  and  congruent. 


146  SOLID   OR   SPATIAL   GEOMETRY. 

The  volume  described   by  these  triangles   in  revolving 
about  OL  as  axis,  p  being  the  common  apothem,  is 

|i)2^(pr.  of  ^5  on  OL  +  ^v.  oi  BC on  OL +  ...). 
But  at  the  limit  when  the  number  of  points  A,  B,  C ... 
is  indefinitely  increased,  and  the  distance  between  them 
is  correspondingly  diminished,  the  generating  figure  be- 
comes the  sector  of  a  circle,  p  becomes  the  radius,  and 
the  sum  of  the  projections  of  the  bases  of  the  triangles 
is  the  projection  of  the  arc,  and  the  figure  generated  is 
a  sector  of  a  sphere. 

Therefore,  the  volume  of  a  sector  of  a  sphere  =  |  tt?-^  x 
pr.  of  the  generating  arc  on  the  axis. 

Cor.  If  the  generating  arc  forms  a  semicircle,  its 
projection  on  the  axis  is  2r,  and  the  figure  generated 
is  a  sphere.        .    ^^j  ^^  ^  ^^^^^^  ^  4  ^^. 

EXERCISES  K. 

1.  Solve  Ex.  6  of  Set  I.,  by  the  principle  of  153. 

2.  AX  is  an  axial  line,  and  PM  is  a  perpendicular  to  this 
line  from  a  point  P  on  a  curve  which  starts  from  A.  If  PM^ 
=  cAM,  where  c  is  a  constant,  show  that  the  volume  described  by 
one  revolution  about  AX,  is  one-half  that  of  the  circumscribing 
cylinder. 

3.  The  volumes  of  the  circumscribing  cylinder,  the  sphere,  and 
the  cone  with  the  same  base  and  altitude  as  the  cylinder,  are  as 
the  numbers  3,  2,  and  1. 

4.  The  volumes  of  the  cylinder  circumscribing  a  semisphere, 
the  semisphere,  and  the  cone  with  base  and  altitude  of  the  cylin- 
der, are  as  the  numbers  3,  2,  1. 

6.  A  plane  cuts  a  sphere,  and  its  circumscribed  cylinder  parallel 
to  the  base  ;  then  twice  the  volume  of  the  segment  is  equal  to  the 
intercepted  volume  of  the  cylinder  and  twice  the  volume  of  the 
sphere  on  the  altitude  of  the  segment  as  diameter. 


THEOREM  OF  PAPPUS.  147 

C.    Theorem  op  Pappus  or  Guldinus  for 
Volumes. 

155.  The  mean  centre  of  a  system  of  complanar  points 
for  a  system  of  multiples  is  defined  (P.  Art.  240)  as  the 
point  of  intersection  of  two  lines,  L  and  M,  for  which 

2(a  .  AL)  =  0,  and  2(a  •  AM)  =  0; 

where  ^  is  a  representative  point,  AL  and  AM  represen- 
tative perpendiculars  from  ^  to  X  and  M  respectively, 
and  a  a  representative  weight  or  number. 

Also  (P.  Art.  241),  if  0  be  the  mean  centre  of  the 
system,  and  L  be  any  line  complanar  with  the  system, 

'^{a-AL)  =  -^{a)OL. 

We  have  to  deal  here  with  the  mean  centre  of  the 
area  of  a  figure,  and  later  on  with  the  mean  centre  of 
the  perimeter  of  a  figure. 

156.  When  a  plane  figure  has  an  axis  of  symmetry, 
the  mean  centre  of  the  figure  lies  on  this  axis. 

For  every  point  in  the  area  upon  one  side  of  the  axis 
of  symmetry  there  is  a  point  upon  the  other  side  exactly 
corresponding  in  every  respect.  So  that  if  L  be  the 
axis  of  symmetry  and  Ai,  A2  be  corresponding  points, 
we  have  ^,L  +  A2L  =  0.  And  since  the  whole  area  is 
represented  by  pairs  of  such  corresponding  elements, 
2(a  •  AL)  =  0,  OT  L  passes  through  the  mean  centre. 

Cor.  1.  AVhen  a  figure  has  two  axes  of  symmetry,  the 
mean  centre  of  area  is  the  point  of  intersection  of  the 
axes. 


148  SOLID   OR   SPATIAL   GEOMETRY. 

This  is  the  case  with  the  square,  the  rectangle,  the 
rhombus,  all  regular  polygons,  the  circle,  and  some  other 
figures. 

157.  If  the  area  of  a  figure  be  supposed  to  be  made 
up  of  elemental  squares,  the  centres  of  these  squares, 
being  their  mean  centre  of  area,  will  represent  the  points, 
A,  B,  C,  etc.,  in  a  system  of  points,  andthe  areas  of  the 
several  squares  will  represent  the  weights. 

But  since  the  squares  are  all  equal,  the  weights  are 
all  equal,  and  may  be  left  out  of  consideration.  With 
this  understanding  we  have  for  the  mean  centre  of  area, 
'^{AL)  =  0,  where  L  passes  through  this  centre;  and 
^{AL)  =  n'  OL,  where  0  is  the  mean  centre,  L  is  any 
line  not  passing  through  0,  and  n  is  the  number  of 
elements  under  consideration. 

158.  Theorem.  The  join  of  the  mean  centres  of  two 
systems  passes  through  the  mean  centre  of  the  system 
composed  of  the  two  taken  together. 

This  theorem  is  almost  self-evident. 

For  if  2  (a'  •  A'L)  =  0  and  2  (a"  •  A"L)  =  0  denote  the 
two  systems,  and  L  is  the  join  of  their  mean  centres,  we 
have  at  once 

:S,{a''A'L-\-a"'A"L)=0; 
which  is  of  the  type  2  (a  •  AL)  =  0. 

Cor.  If  any  number  of  systems  have  their  mean 
centres  collinear,  the  mean  centre  of  the  system  com- 
posed of  all  taken  together  lies  on  the  line  of  coUinearity. 

159.  Theorem.  The  mean  centre  of  a  parallelogram  is 
its  geometric  centre,  i.e.  the  intersection  of  its  diagonals. 


THEOREM  OF   PAPPUS. 


149 


Let  ABCD  and  EFOH  be  two  congruent  parallelo- 
grams, superposable  with  E  on  A,  F  on  B,  G  on  C,  and 
H  on  D.  Their  mean  centres  of  area  are  then  coinci- 
dent. But  the  parallelograms  are  also  superposable  with 
E  on  C,  F  on  D,  G  on  A,  and  H  on  B;  and  their  mean 
centres  of  area  are  again  coincident. 

Hence  the  mean  centre  of  each  is  the  geometric  centre. 

Cor.  In  like  manner  it  may  be  shown  that  when  any 
figure  has  a  geometric  centre,  that  centre  is  also  the 
mean  centre  of  its  area. 

160.  Mean  centre  of  the  area  of  a  triangle.  Let  BD 
be  a  median  to  the  triangle  ABO.  Draw  EF  and  GH 
two  near  lines  each  parallel  to  AC,  and  draw  EI,  FJ 
parallel  to  BD. 

The  parallelogram  EJ'\^  an  element 
of  the  area  of  the  triangle,  and  the 
sum  of  the  areas  of  these  elements, 
when  taken  at  the  limit,  is  the  area 
of  the  triangle. 

But  as  BD  bisects  EF  and  IJ,  it 
passes  through  the  mean  centres  of  all 
the  elements  of  which  EJ  is  a  type. 

Therefore  »(Art.  158),  the  centre  of  area  of  the  triangle 
lies  on  BD;  and  as  it  lies  on  both  of  the  other  medians, 
the  centre  of  area  of  a  triangle  is  its  centroid. 

Def.  On  account  of  the  foregoing,  we  shall  call  the 
mean  centre  of  area  of  any  figure  its  centroid. 

161.  Theorem.  The  orthogonal  projection  of  the  mean 
centre  of  any  complanar  system  is  the  mean  centre  of 
the  projection  of  the  system  for  the  same  multiples. 


150  SOLID   OR   SPATIAL  GEOMETRY. 

Let  A,  B,  C,  '••  be  the  elements  in  the  plane  U,  and 
A',  B',  C,  '"  be  their  projections  on  the  plane  V.  Take 
L,  any  line  in  U,  through  the  mean  centre  0,  and  let  L' 
and  0'  be  the  projection  of  L  and  0  on  V. 

Then  2  (a  •  AL)  =  0.  (P.  Art.  240.) 

But  AL,  BL,  CL,  etc.,  are  all  parallel,  and  A'L',  B'L', 
C'L',  etc.,  are  all  parallel.     Therefore,  the 

Z  {AL  •  A'L')  =  Z  (BL  ■  B'L')  =  etc., 
and  hence 

AL       BL 

^'  =  ^=-=^'^^^' 

.-.  ^{a-AL)  =  0=p^{a' A'L'), 

or  2(a.^'i')  =  0; 

and  L'  passes  through  the  mean  centre  of  the  projected 
system.  And  as  this  is  true  for  all  directions  of  L  and 
L'  in  their  respective  planes,  0'  is  the  mean  centre  of 
the  projected  system ;  or  the  projection  of  the  mean 
centre  of  the  system  in  U  is  the  mean  centre  of  the  pro- 
jected system  in  V. 

162.  Def.  Let  us  call,  in  general,  a  figure  of  the  type 
of  the  cylinder  or  prism,  but  with  non-parallel  bases,  a 
cylindroid. 

Suppose  a  system  of  near  equidistant  planes  parallel 
to  the  axis  to  cut  the  cylindroid.  These  divide  it  into 
laminae  parallel  to  the  axis.  Now  suppose  a  second  set 
of  planes,  parallel  to  the  axis,  to  cut  the  first  system  at 
right  angles,  and  to  have  the  distance  between  consecu- 
tive planes  the  same  as  in  the  first  system. 


THEOREM   OF   PAPPUS. 


151 


These  planes  divide  the  cylindroid  into  elementary- 
prisms  on  square  bases.  These  form  the  prismatic  ele- 
ments of  the  figure,  and  the  sum  of  their  volumes,  at  the 
limit,  as  their  bases  are  indefinitely  diminished  and  their 
number  is  correspondingly  increased,  is  the  volume  of 
the  cylindroid. 

163.  AGB  is  a  cylindroid  having  the  base  ABH  nor- 
mal to  the  axis,  and  the  base  CGD  oblique  to  the  axis. 

Let  PQ  be  a  prismatic  element, 
the  area  of  whose  base  is  y8,  and  let 
the  line  CD  be  taken  parallel  to  the 
common  line  of  the  planes  of  the 
bases,  and  let  AB  be  the  orthogonal 
projection  of  CD  on  the  lower  base. 

Draw  QFA.  to  AB,  and  FE  nor- 
mal to  the  base  ABH.     Then  FE  h. 
is  parallel  to  QP,  and  meets  CD  in 
some  point  E.     Draw  ER 1.  to  PQ. 
Join  EP.     Then  EP  is  ±  to  CD,  and  ERQF  is  a  rect- 
angle, and  EF=  QE. 

The  volume  of  the  prismatic  element  PQ  is 

l3'PQ  =  l3-PB-\-p'BQ  =  ^-PB  +  l3'EF. 

And  since  the  bases  of  all  the  prismatic  elements  have 
the  same  area,  and  EF  is  constant,  the  sura  of  the 
volumes  is 

^{l3-PR)-\-'${f3)EF. 

But  '^{/3)EF is,  at  the  limit,  the  volume  of  the  cylin- 
der UK,  whose  base  is  ABH,  and  altitude  EF. 

In  order  that  this  may  be  equal  to  the  cylindroid,  we 
must  have  2(/?  •  PE)=0 ;  and  as  every  element  PM  is  in 


152  SOLID   OR   SPATIAL   GEOMETRY. 

a  constant  ratio  to  the  corresponding  element  EP,  and 
P  is  constant,  we  must  have  %{EP)  =  0. 

Or  CD  must  pass  through  the  centroid  of  the  upper 
base,  CGD,  of  the  cylindroid;  and  (Art.  161)  AB 
passes  through  the  centroid  of  the  lower  base. 

Hence,  however  the  directions  of  the  planes  of  section 
which  give  the  bases  may  vary,  provided  they  do  not 
meet  within  the  limits  of  the  cylindroid,  the  volume 
remains  unchanged,  while  the  distance  between  the 
centroids  of  the  bases  remains  the  same. 

Cor.  The  volume  of  a  cylindroid  is  the  area  of  a  right 
section  multiplied  by  the  distance  between  the  centroids 
of  the  bases. 

164.  Let  the  plane  figure  X,  invariable  in  form  and 
dimensions,  move  from  a  position  AB  to  another  posi- 
tion CD,  in  such  a  manner  that  its 

direction  of  motion,  whether  follow-  b,,,---'-'7^7'\ ~~^ 

ing   a  line  or  a   curve,  is   always  UX      (-*V  r L    1 

normal  to  its  plane.  \,*o \     \  i  y      (  "/ 

Take  two  near  positions  of  X  as  \i--^HK  c 
at  GH  and  JK,  and  consider  these 
as  bases  of  a  cylindroid  forming  an  element  of  the  figure 
generated  by  the  motion  of  X.  If  P  and  Q  be  the 
centroids  of  the  bases,  the  volume  of  the  elementary 
cylindroid,  GK,  is  X  •  PQ,  where  X  is  the  area  of  the 
generating  figure. 

And  at  the  limit,  when  P  approaches  indefinitely  to  Q, 
the  sum  of  the  cylindroids  is  the  generated  spatial  figure, 
and  the  sum  of  the  elements  PQ  is  the  path  of  the 
centroid  of  X. 


THEOREM  OF   PAPPUS.  153 

Therefore,  when  a  plane  figure,  invariable  in  form  and 
dimensions,  moves  in  a  path  which  is  at  every  instant 
normal  to  the  plane  of  the  figure,  the  whole  volume 
described  is  the  area  of  the  figure  multiplied  by  the 
length  of  path  moved  over  by  the  centroid  of  the  figure. 

This  is  the  statement  of  the  theorem  as  first  given  by 
Pappus  (about  300),  and  afterwards  reproduced  by 
Guldinus  (1577-1643),  and  usually  called  after  his  name. 

Cor.  When  a  plane  figure  revolves  about  a  complanar 
axis,  the  direction  of  motion  of  the  centroid  is  at  all 
times  necessarily  normal  to  the  plane  of  the  figure,  and 
the  volume  described  in  one  revolution  is  the  area  of  the 
plane  figure  multiplied  by  the  circumference  traced  by 
its  centroid. 

Ex.  A  circle  revolves  about  a  complanar  line  lying 
without  it ;  the  figure  generated  is  called  an  anchor  ring. 
To  find  its  volume. 

Let  r  be  the  radius  of  the  generating  circle,  and  R  be 
the  distance  of  its  centre  from  the  axis.     Then 

vol.  =27ri2.7rr2  =  27rVi2. 

EXERCISES  L. 

1.  Find  the  position  of  the  centroid  of  a  semicircle. 

2.  The  circle  which  generates  an  anchor  ring  is  divided  by  a 
diameter  parallel  to  the  axis ;  compare  the  volume  described  by 
the  outer  and  the  inner  half  of  the  circle. 

3.  A  semicircle  revolves  about  its  limiting  diameter.  Any 
segment  whose  chord  is  parallel  to  the  axial  line  describes  a 
volume  equal  to  that  of  a  sphere  on  the  chord  of  the  segment  as 
diameter. 


154  SOLID   OR   SPATIAL   GEOMETRY. 


4.  The  distance  from  the  centre  of  a  circle  to  the  centroid  of 

2  c* 

any  segment,  is ,  where  c  is  the  half  chord  of  the  segment,  and 

3  S 
S  is  its  area. 

5.  The  distance  of  the  centroid  of  a  segment  from  its  chord  is 

2c3     c'2  , 

where  c'  is  the  chord  of  half  the  arc,  and  v  is  the  versed  sine  of 
the  arc  (P.  Art.  176.  Cor.  1), 

6.  An  arc  of  a  circle  revolves  about  its  chord ;  the  volume 
generated  is 

The  figure  generated  is  called  a  circular  spindle. 

7.  The  centroid  of  a  semicircle  is  at  the  distance  —  from  the 
centre  of  the  circle.  ^ 

8.  A  semicircle  revolves  about  a  tangent  at  its  middle  point. 
The  volume  described  is 

i7rr3(3  7r-4). 

9.  A  square  with  side  s  revolves  about  a  line  through  one 
vertex,  making  an  angle  d  with  a  side,  and  not  crossing  the  square. 
The  volume  described  is  7rs^(sin  $  +  cos  6). 

10.   A  plane  cuts  through  a  right  circular  cylinder  so  as  to  cut 
one  base  only.    The  volume  of  the  portion  removed  is 

-{lc'-ir-v)S}; 

V 

where  h  is  the  height  of  the  convex  part,  r  is  the  radius  of  the 
cylinder,  and  v,  c,  and  S  denote  the  versed  sine,  semichord,  and 
area  of  the  segment  of  the  base. 

This  figure  is  called  an  ungula  of  a  right  circular  cylinder. 


SECTION  4. 

Planimetry — The  Measurement  of  the  Areas 
OF  Surfaces,  or  Superficies. 

165.  When  a  spatial  figure  is  bounded  by  plane  faces 
only,  the  area  of  its  surface  is  the  sum  of  the  areas  of 
its  faces. 

For  such  figures  no  special  method  is  required  outside 
of  the  processes  of  plane  geometry. 

The  area  of  a  curved  surface  is  usually  derived  from 
that  of  a  polyhedron  by  going  to  the  limit,  and  suppos- 
ing the  number  of  polyhedral  faces  to  be  indefinitely 
increased  while  the  size  of  each  face  is  correspondingly 
diminished. 

In  some  curved  surfaces,  however,  we  may  suppose 
the  surface  to  be  brought  to  coincide  with  a  plane  by  a 
sort  of  unrolling  of  the  surface  without  stretching  or 
distorting  it  in  any  of  its  parts.  Such  surfaces  are  said 
to  be  developable  ;  and  when  the  surface  is  brought  to 
coincide  with  a  plane,  it  is  said  to  be  developed  on  the 
plane. 

Thus  a  sheet  of  paper  may  be  rolled  into  a  cone  or  a 
cylinder,  but  it  cannot  be  bent  into  a  sphere. 

The  cylinder  and  the  cone  are  accordingly  developable 
surfaces,  while  the  sphere  is  not. 

It  is  readily  seen  that  none  but  ruled  surfaces  can  be 
developable.    Kuled  surfaces  are  not,  however,  all  devel- 

155 


156  SOLID   OR   SPATIAL   GEOMETRY. 

opable,  and  those  which  are  not  so  are  called  skew  sur- 
faces. 

166.   Development  of  the  conical  surface. 

Let  0  be  the  centre  of  a  circular  cone,  and  L  he  a, 
generating  line. 

On  L  take  any  point,  P,  and  through  P  draw  the 
cone-circle  APB  with  0  as  vertex. 


With  any  point,  Q,  as  centre,  and  QD  =  OP  as  radius, 
describe  an  arc,  DE,  equal  in  length  to  the  circumfer- 
ence of  the  circle  APB. 

The  figure  QDE,  a  sector  of  a  circle,  is  the  develop- 
ment of  the  conical  surface  lying  between  the  centre  0 
and  the  cone-circle  APB. 

It  must  be  remarked  that  the  construction  here  given 
is  theoretical  only,  since  we  have  no  method  in  elemen- 
tary'- geometry  of  constructing  an  arc  of  one  circle  equal 
in  length  to  a  given  arc  of  another  circle,  when  the 
circles  have  different  and  incommensurable  radii.  This 
difficulty  will  not,  however,  vitiate  any  application  to  be 
made  of  this  principle. 


PLANIMETRY.  157 

167.  Area  of  surface  of  right  circular  cone. 

For  the  closed  cone  0  •  APB  it  is  evident  that  the 
area  of  the  curved  surface  is  equal  to  the  area  of  its 
development,  i.e.  of  the  circular  sector  QDE,  and  this 
is  one-half  the  length  of  the  arc  DE  multiplied  by  the 
radius  QD. 

But  the  arc  DE  is  equal  in  length  to  the  circle  APB, 
and  the  radius  QD  is  equal  to  OP. 

Therefore,  denoting  the  circumference  of  the  base  by 
C,  and  the  slant  height,  AO  or  BO,  by  S, 

curved  surface  =  \  CS. 

168.  Frustum  of  a  right  circular  cone. 

Drawing  a  second  cone-circle,  aph,  to  the  vertex  0, 
and  the  development  Qde,  we  have  for  the  frustum, 

area  =  sector  QDE  —  sector  Qde. 

Or,  denoting  the  circumference  of  apb  by  c, 

2area=  OP-C-Op-c. 

But  OP  =  Pp  +  0p:  and  ^=  - : 

Op      c 

.-.  ^  =  ^^^,  (P.  Art.  195.  1.) 

Op         c    '  ^  ' 

and  2  area  =  Pi?.  C-f  Op (C—c), 

or  area  of  surface  =  \Pp  {C -\-g). 

169.  In  the  cylinder,  0,  and  therefore  Q,  goes  to  infin- 
ity, and  QD  and  QE  become  parallel. 

Hence  DE  and  de  become  equal  and  parallel  lines,  and 
the  development  DdeE  is  a  rectangle. 


158 


SOLID   OR   SPATIAL   GEOMETRY. 


Or,  if  h  be  the  height  of  the  cylinder,  and  r  be  the 
radius  of  the  base, 

convex  surface  =  2  irrh. 

170.    Area  of  the  surface  of  a  sphere. 

Let  AB  be  a  quadrant  of  a  circle  which  generates  a 
semisphere   by    revolving   about 
OB  as  an  axis. 

Take  two  near  points  on  the 
curve,  P  and  Q,  which  at  the 
limit  come  into  coincidence,  and 
draw  the  chord  PQ.  This  chord 
describes  the  convex  surface  of 
a  frustum  of  a  cone,  and  the  area 
of  the  surface  is  o     v  r   q 


A        D  E  c 

/  J 


where  Pp  and  Qq  are  _Ls  upon  OB. 


(Art.  168.) 


Take  R,  the  middle  point  of  PQ,  and  draw  Rr  A.  to  OB 
and  join  RO,  and  also  draw  QS 1.  to  Pp. 
Then  the  surface  described  by  PQ  is 

2Tr 'PQ'Rr. 

And  on  account  of  the  similar  A  PQS  and  ORr,  the 
surface  described  by  PQ  is 

27r-  OR-pq. 

And  the  convex  surface  described  by  a  system  of  chords, 
forming  the  sides  of  a  regular  polygonj  is 


27r-  OR--S.(pq). 


PLANIMETRY.  159 

But  at  the  limit  when  P  comes  to  Q,  the  apothem,  OR, 
becomes  the  radius,  and  the  polygon  becomes  the  circle ; 
and  the  surface  described  by  any  arc,  PQ,  is 

2  7rr  X  proj.  of  the  arc  on  the  axis. 

Now,  2  ttt  is  the  circumference  traced  by  D,  a  point 
on  the  circumscribed  rectangle  ACBO,  and  the  projection 
of  the  arc  is  equal  to  DE ; 

Therefore,  the  convex  surface  described  by  the  arc  PQ 
is  equal  to  the  convex  surface  described  by  DE. 

Hence,  if  a  sphere  and  its  circumscribed  right  cylinder 
be  cut  by  two  planes  parallel  to  the  bases  of  the  cylinder, 
the  area  of  the  curved  surface  intercepted  between  the 
planes  is  the  same  for  the  sphere  as  for  the  cylinder. 

Cor.  The  area  of  the  surface  of  a  sphere  is  equal  to 
that  of  the  curved  surface  of  its  circumscribed  right 
cylinder. 

Therefore,  the  area  of  the  surface  of  a  sphere  is 

47rr2, 

or  four  times  the  area  of  a  great  circle. 

171.  We  may  consider  a  sphere  as  circumscribing  a 
conspheric  polyhedron  with  an  indefinite  number  of  very 
small  faces.  Considering  these  faces  as  bases  of  pyra- 
mids having  their  vertices  in  common  at  the  centre  of 
the  sphere,  the  sum  of  these  pyramids  at  the  limit,  when 
the  number  is  indefinitely  increased,  and  the  size  of  each 
base  is  correspondingly  diminished,  is  the  volume  of 
the  sphere,  and  the  sum  of  their  bases  is  the  surface 
of  the  sphere. 


160  SOLID   OR   SPATIAL   GEOMETRY. 

But  each  pyramid  is  \Bh,  and  their  sum  is  ^2(-S/i). 
Or  writing  S  for  %B,  and  r  for  h, 

where  V  is  the  volume  of  the  sphere,  and  S  is  the  area 
of  the  surface. 


Theorem  of  Pappus  or  Guldinus  for 
Surfaces. 

172.  For  convenience  we  shall  call  the  mean  centre 
of  the  perimeter  of  a  plane  figure  its  centre  of  figure. 

The  general  theorems  respecting  the  mean  centre  as 
developed  in  Arts.  158  and  160  apply  to  the  centre  of 
figure  in  the  same  manner  as  to  the  centre  of  area. 

173.  Two  equal  line-segments,  AB  and  CD,  are  con- 
gruent whether  A  is  placed  on  G,  and  B  on  D,  or  A  on  D, 
and  B  on  C,  and  hence  the  centre  of  figure  of  a  line- 
segment  is  its  middle  point. 

Then,  in  any  rectilinear  figure  which  has  an  axis  of 
symmetry,  the  sides  exist  in  congruent  pairs  which  are 
symmetrically  disposed  upon  opposite  sides  of  the  axis 
of  symmetry. 

And  hence  if  A  and  A'  denote  the  middle  points  of 
two  sides  forming  a  symmetrical  pair,  AL=  —A'L,  or 
AL  -\-  A'L  =  0 ;  where  L  is  the  axis  of  symmetry. 

Therefore,  2  {AL)  =  0 ;  ox  L  passes  through  the  cen- 
tre of  figure. 

Hence,  when  a  rectilinear  figure  has  an  axis  of  sym- 
metry, the  centre  of  figure  lies  upon  that  axis. 


THEOREM  OP  PAPPUS. 


161 


/ 

\^^ 

'/ 

E 

\ 

c 

R--' 

V 

R 

\ 

Q 

y 

^^■^~.__£_ 

\ 

. 

T 

Cor.  When  a  rectilinear  figure  has  two  axes  of  sym- 
metry, their  point  of  intersection  is  the  centre  of 
figure. 

174.  Let  AGB  be  a  cylindroid  having  the  base  ABH 
normal  to  the  axis,  and  the  base  CGD  oblique  to  it. 
Suppose  the  convex  surface  to  be 
divided  in  very  narrow  strips  of 
equal  width  throughout,  and  paral- 
lel to  the  axis  of  the  cylindroid  and 
equal  in  width  to  one  another ;  and 
let  b  denote  the  breadth  of  one  of 
these  elements  of  surface,  and  let 
ST  denote  the  line  along  the  middle  hI-' 
of  the  strips.  Then  ST  is  normal 
to  the  base  ABH.  *" 

Let  CD  be  parallel  to  the  common  line  of  the  planes 
of  the  bases,  and  let  AB  be  the  projection  of  CD  on  the 
lower  base.  Draw  TF  ±  to  AB  and  FE  ±  to  CD.  Also 
draw  ^F±  to  ST,  and  join  ES. 

Then,  EVTF  is  a  rectangle,  and  VT=  EF. 

The  area  of  the  element  represented  by  ST  is 

b-ST,  01  b-  EF+  b  ■  SV. 

And  the  sum  of  these  elements,  of  which  the  one 
represented  by  /ST  is  the  type,  is, 

^{b'EF)+'^(b'SV). 

But  EF  is  constant,  and  2(&)  is  the  circumference  of 
the  base  ABH. 

Therefore  ^{b  •  EF)  is  the  convex  surface  of  the 
cylinder,  or  prism,  whose  base  is  ABH,  and  whose  alti- 


162  SOLID   OE   SPATIAL   GEOMETRY. 

tude  is  EF.  And  that  this  may  be  equal  to  the  whole 
convex  surface  we  must  have  2(6  •  SV)  =  0. 

But  as  all  the  elements  have  the  same  width,  h  is 
constant,  and  SV:  SE  is  constant ; 

.-.  ^{ES)  =  0; 

or,  CD  passes  through  the  centre  of  figure  of  the  upper 
base ;  and  hence  AB  passes  through  the  centre  of  figure 
of  the  lower  base. 

Therefore,  the  area  of  the  convex  surface  of  a  cylin- 
droid  is  the  circumference  of  a  right  section  multiplied 
by  the  distance  between  the  centres  of  figure  of  the 
bases. 

175.  Let  the  plane  figure  X,  invariable  in  form  and 
dimensions,  move  with  centre  of  figure  on  the  path 
OPQR,  and  so  that   the  direction 

of  the  path  is  at  all  points  normal  b^ — T^^T'^X tR 

to  the  plane  of  the  figure,  and  let  C\      (.-,  V*  \ f-     I 

G^jEf  and  Jif  be  two  near  positions,   \»q\     \  //      \^J 
which  at  the  limit  come  into  coin-    \J- — hk         c 
cidence. 

The  surface  of  the  elementary  cylindroid  GK  is  the 
circumference  of  X  x  PQ ;  and  the  area  of  the  surface 
of  the  figure  generated  by  the  motion  of  X  is 

%{PQ)  X  circum.  of  X. 

But  2(PQ)  is  the  path  OPQR---,  and  circumference  of 
X  is  constant. 

Therefore,  the  area  of  the  surface  described  by  a  plane 
figure,  invariable  in  form  and .  magnitude,  which  moves 
so  that  its  direction  of  motion  is  at  each  point  normal  to 


THEOREM   OF   PAPPUS.  163 

its  plane,  is  the  circumference  of  the  generating  figure 
multiplied  by  the  length  of  path  described  by  its  centre 
of  figure. 

Cor.  When  a  plane  figure  revolves  about  a  complanar 
line  as  axis,  the  direction  of  motion  is  necessarily  normal 
to  the  plane  of  the  figure,  and  the  surface  described  has 
for  its  area  the  circumference  of  the  figure  multiplied  by 
the  circumference  traced  by  its  centre  of  figure. 

Ex.  To  find  the  surface  of  an  anchor  ring.  The 
centre  of  figure  is  the  centre  of  the  generating  circle, 
and  the  circumference  traced  is  ^irR. 

. '.  area  of  surface  =  2  7rr  •  2  nK  =  4  tt^Hv. 

176.  The  two  theorems  which  go  under  the  name  of 
Guldin's  theorems,  but  which  were  discovered  by  Pappus, 
express  relations  of  the  highest  importance  in  mathe- 
matics both  pure  and  applied.  They  enable  us  to  find 
the  centroid  of  a  generating  figure  when  the  volume  of 
the  generated  figure  is  known,  or  the  centre  of  figure  of 
a  generating  figure  when  the  area  of  the  surface  of  the 
generated  figure  is  known,  and  vice  versa. 

Thus  knowing  the  volume  of  a  sphere,  we  can  readily 
find  the  centroid  of  a  semicircle,  and  knowing  the  sur- 
face of  a  sphere  enables  us  to  find  the  centre  of  figure 
of  a  semicircular  arc. 

EXERCISES  M. 

1.  The  circle  describing  an  anchor  ring  is  divided  by  a  diameter 
parallel  to  the  axis.  Show  that  the  difference  between  the  sur- 
faces described  by  the  outer  and  the  inner  part  is  eight  times  the 
area  of  the  generating  circle. 


164  SOLID  OR   SPATIAL   GEOMETRY. 

2.  The  convex  surface  of  a  cone  is  irrs ;  and  the  entire  surface 
is  irr  (r  +  s) ;  where  s  is  the  slant  height,  and  r  is  the  radius  of 
the  base. 

3.  The  entire  surface  of  a  conical  frustum  is 

4.  The  areas  of  the  surfaces  of  the  regular  polyhedra  are 
as  follows : 

Tetrahedron,  e^-^S;  Cube,  Ge^;  Octahedron,  e-2^S;  Dodeca. 
hedron,  enb^{l  +  ly/5)  ;  Icosahedron,  e^5^S. 

5.  The  distance  between  the  centre  of  a  circle  and  the  centre 

of  figure  of  any  arc  of  the  circle  is »  where  I  is  the  length  of 

the  arc,  and  c  and  r  denote  as  usual. 

6.  The  area  of  the  surface  of  a  circular  spindle  Is 

-|2cc'2-(c2-«2)  A. 

7.  The  convex  surface  of  an  ungula  of  a  right  circular  cylinder  is 


-(2cr+r-v-l).    See  Ex.  10.  K. 

V 


Part  IY. 

PROJECTIONS  AND  SECTIONS. 

SECTION  1. 
Perspective  Projection. 

177.  Def.  Let  P  be  a  variable  point  on  a  plane  figure 
X;  let  O  be  a  fixed  point  not  complanar  with  X\  and 
let  L  be  the  line  OP. 

When  P  describes  the  figure  X,  L  describes  the  per- 
spective projection  of  X  in  space,  or  the  spatial  projec- 
tion of  X,  and  0  is  the  centre  of  the  projection. 

If  the  spatial  projection  be  cut  by  a  plane  V,  the 
figure  of  section  is  a  plane  figure  called  the  perspective 
projection  of  X  on  V  for  the  centre  0. 

When  0  goes  to  infinity,  L  has  a  fixed  direction,  and 
is  parallel  to  a  fixed  line  for  all  positions  of  P,  and  the 
projection  becomes  parallel  projection. 

If  the  direction  of  0  at  infinity  is  normal  to  V,  the 
projection  on  V  becomes  orthogonal  or  ortJiographic  pro- 
jection, which  is  thus  a  special  case  of  perspective  pro- 
jection. 

If  the  direction  of  0  at  infinity  is  perpendicular  to 
165 


166  SOLID   OR   SPATIAL   GEOMETRY. 

the  plane  of  X,  but  oblique  to  V,  the  projection  may  be 
called  the  ant-orthogonal  projection  of  X  on  V. 

In  what  follows  in  this  section,  projection  will  mean 
projection  with  0  finite  unless  otherwise  stated,  and  the 
projection  will  mean  the  figure  of  section, 

178.  We  observe  that  in  a  way  projection  and  section 
are  reciprocal  processes,  as  by  projecting  a  plane  figure 
we  get  a  spatial  one,  and  by  cutting  the  spatial  one  by 
a  plane  we  return  to  a  plane  figure. 

And  this  passing  from  one  plane  figure  to  another 
through  a  spatial  figure  may  be  repeated  as  often  as  we 
please. 

179.  Since  the  generator  L  is  unlimited,  the  spatial 
figure  extends  to  infinity  on  both  sides  of  the  centre,  and 
admits  of  section  on  either  side  or  section  on  both  sides 
by  the  same  plane,  examples  of  which  will  occur  here- 
after. 

180.  The  following  theorems  are  fundamental  : 

1.  A  line  projects  into  a  line. 

For  the  spatial  projection  of  a  line  is  a  plane,  and 
every  plane  section  of  a  plane  is  a  line. 

2.  The  point  of  intersection  of  two  lines  projects  into 
the  point  of  intersection  of  the  projections  of  the  lines. 

Hence  the  projection  of  a  plane  rectilinear  figure  is  a 
plane  rectilinear  figure  having  the  same  number  of  sides 
and  vertices  as  the  projected  figure. 

3.  A  curve  projects  into  a  curve,  and  a  tangent  to  the 
curve  into  a  tangent  to  its  projection. 


PERSPECTIVE   PROJECTION. 


167 


For  the  coincident  points  common  to  the  curve  and 
the  tangent  become  coincident  points  common  to  the 
curve  and  the  tangent  in  the  projection. 

On  account  of  its  character,  perspective  projection  is 
also  called  conical  projection. 


181.  Let  0  be  a  centre  of  projection,  and  let  ?7and  V 
be  two  planes,  neither  of  which  contains  0,  and  whose 
common  line  is  AB. 

Through  0  pass  a  plane  parallel  to  V,  and  let  it  meet 
Uin  I.  Then  /  is  parallel  to  AB.  Take  P,  any  point 
in  /,  and  let  OP  be  the  axis  of  an  axial  pencil.  The 
section  of  this  pencil  by  [/"is  the  flat  pencil  P  •  QES,  ••• ; 
and  since  OP  meets  V  at  infinity,  the  section  of  the 
axial  pencil  by  F  is  a  set  of  parallels,  QQ',  BR',  SS', 
etc.,  ..•  (Art.  21.  Cor.  2). 


168  SOLID   OR   SPATIAL   GEOMETRY. 

Hence  the  flat  pencil  on  C/'with  vertex  at  P  projects 
into  a  parallel  system  on  V,  and  the  direction  of  the 
system  is  that  of  the  axis  OP. 

Def.  P  is  the  vanishing  point  on  U  for  the  parallel 
system  on  V,  or  it  is  a  vanishing  point  for  V,  i.e.  for 
some  system  of  parallels  on  V;  and  /,  which  is  the  locus 
of  vanishing  points  for  different  systems  of  parallels  on 
V,  is  the  vanishing  line  on  U,  or  the  vanishing  line  for  V. 

Similarly,  by  passing  a  plane  through  0  parallel  to  U, 
we  obtain  the  line  J,  on  V,  as  the  vanishing  line  for  U. 

Thus  either  plane  may  be  taken  indifferently  as  the 
plane  of  the  figure,  and  the  other  plane  becomes  the 
plane  of  the  projection,  or  the  plane  of  section.  The 
operation  of  projection  is  thus  completely  reversible. 

Cor.  1.  Any  point  is  projected  to  infinity  by  taking 
the  plane  of  section  parallel  to  the  join  of  that  point 
with  the  centre  of  projection. 

Cor.  2.  Any  line  is  projected  to  infinity  by  taking 
the  plane  of  section  such  that  the  given  line  may  be  the 
"vanishing  line  for  that  plane. 

Cor.  3.  If  P  goes  to  infinity  along  /,  OP  becomes 
parallel  to  AB.  But  the  flat  pencil  whose  vertex  is  at 
infinity  is  a  set  of  parallels. 

Therefore,  lines  parallel  to  the  common  line  of  the 
planes  project  into  lines  having  the  same  direction,  i.e. 
into  a  set  of  parallels. 

Cor.  4.  If  the  planes  U  and  V  are  parallel,  the  figure 
and  its  projection  are  similar  (Art.  28.  Cor.  2),  and 
parallel  lines  project  into  parallel  lines. 


PERSPECTIVE  PROJECTION. 


169 


182.  Application.     To  prove  that  the  middle  points  of 
the  three  diagonals  of  a  tetragram  are  collinear. 

ABGD  is  a  tetragram,  and 
AG,  BD,  EF  are  its  three 
diagonals. 

Let  BD  and  FE  produced 
meet  in  S,  and  project  S  to 
infinity  in  the  direction  BD. 
Then  BD  and  FE  become 
parallel,  and  AC  is  a  me- 
dian to  the  new  triangle 
AEF  (P.  Ex.  11,  p.  170), 

and  P,  Q,  and  M  are  points  on  this  median,  and  are 
therefore  collinear. 

But  by  projection  a  line  can  come  only  from  a  line. 
Therefore,  P,  Q,  R,  are  always  collinear. 

183.  Theorem.  Anhar- 
monic  relations  are  un- 
changed by  projection. 

Let  ABCD  be  a  range 
on  the  plane  U,  and  let 
A'B'C'D'  be  its  projection 
on  V  from  the  centre  0. 

ABCD,  A'B'C'D',  and  0 
lie  in  one  plane,  and  the  theorem  is  reduced  to  that  of  a 
flat  pencil. 

But  0\ABCDI  =  0{A'B'C'D'l.     (P.  Art.  304.  Cor.  3.) 

.-.   {ABCDl  =  \  A'B'C'D' l 

Or  the  anharmonic   ratio   of  ABCD  is   imchanged  by 
projection.  . 


-y 


170  SOLID   OR   SPATIAL   GEOMETRY. 

Cor.  1.  Any  given  range  can  be  considered  as  having 
come  from  some  other  range  having  the  same  anharmonic 
ratio ;  and  any  range  and  its  projection  are  homographic. 

Cor.  2.  If  ABCD  be  a  harmonic  range,  and  D  be 
projected  to  infinity,  the  projection  of  B  bisects  the  join 
of  the  projections  of  A  and  C. 

184.  Application.  To  show  that  a  diagonal  of  a  tetra- 
gram  is  divided  harmonically  by  the  other  diagonals. 

In  the  figure  of  Art.  182,  project  the  diagonal  EF  to 
infinity  (Art.  181.  Cor.  2),  E  going  in  the  direction  AD, 
and  F  in  the  direction  AB. 

Then  ABCD  becomes  a  parallelogram,  and  0  is  the 
middle  point  of  the  diagonals,  and  therefore  the  middle 
point  of  AC.  But  since  G  goes  to  infinity,  AOCG  is  a 
harmonic  range. 

Therefore,  AOCG  is  always  a  harmonic  range.  Similar 
proofs  may  be  obtained  for  the  other  diagonals. 

185.  Theorem.  Any  angle  less  than  a  straight  angle 
may  be  projected  into  any  required  angle  less  than  a 
straight  angle. 

Let  APC  be  the  given  angle  lying  in  the  plane  U,  and 
let  A  and  C  be  the  points  where  its  arms  meet  the 
plane  of  projection  V. 

With  AC  as  chord  describe  on  Fa  segment  of  a  circle 
ABC  which  shall  contain  the  required  angle,  and  through 
B,  any  point  on  this  segment,  draw  BP.  The  centre  of 
projection  is  at  any  point  on  the  line  BP,  as  at  0. 

For  the  planes  BPA  and  BPC  contain  the  given  angle 
APC  on  U,  and  the  required  angle  ABC  on  V- 


PERSPECTIVE  PROJECTION. 


171 


Cor.  1.  Since  B  is  any  point  on  a  circle,  and  0  is 
any  point  on  the  line  BP,  0  lies  on  the  surface  of  a 
cone  whose  centre  is  P,  and  ^ -,^ 

of  which  OPB  is  a  gener-  \~"-r 

ating  line,  and  the   circle 
ABC  the  director  curve. 

Cor.  2.  Let  /  be  any 
line  in  U. 

Through  /  draw  any 
plane,  W,  and  take  V  par- 
allel to  W.  From  any  point 
on  W,  I  is  projected  to  in- 
finity on  V.  But  W  cuts 
the  cone  in  a  curve,  QOR, 
and  any  point  in  this  curve 
lies  at  the  same  time  upon 
the  surface  of  the  cone  and  upon  the  plane  W. 

Therefore,  from  any  point  in  this  curve  the  Z  APC  is 
projected  into  a  given  angle  ABC,  and  any  line  /  is 
projected  to  infinity. 

Hence  any  number  of  points  can  be  found  all  lying  on 
the  plane  section  of  a  cone,  from  which  as  centre  a  given 
angle  of  a  plane  figure  may  be  projected  into  a  required 
angle,  and  a  given  line  of  the  same  figure  be  projected 
to  infinity. 

186.  Any  quadrilateral  may  be  projected  into  a  rect- 
angle. 

For  if  we  project  the  external  diagonal  to  infinity,  and 
an  angle  of  the  quadrilateral  into  a  right  angle,  the 
figure  becomes  a  parallelogram  with  one  right  angle, 
i.e.  a  rectangle. 


172 


SOLID   OR   SPATIAL   GEOMETRY. 


187.  Theorem.  Any  line-segment  may  be  so  projected 
that  any  point  on  the  line  of  the  segment  may  become 
the  middle  point  of  the  projection. 

1.  Let  the  point  divide  the  segment  internally. 

Let  AB  be  a  given  segment  and  C  be  an  internal  point 
in  it.  Take  D,  the  harmonic  conjugate  to  C  (P.  Art. 
309),  and  taking  any 
point  0  as  centre  of  pro- 
jection, join  OD,  and 
project  the  range  upon  a 
line,  L,  parallel  to  OD. 

Then  since  D  goes  to 
infinity,  C"  bisects  the  seg- 
ment A'B'. 

2.  Let  the  point  D 
divide  the  segment  exter- 
nally. Take  C,  a  har- 
monic conjugate  to  the 
given  point  D,  and  join- 
ing OC,  project  the  range 
upon  any  line  M  parallel 
to  00. 

Then,  since  C  goes  to  infinity,  D"  bisects  the  segment 
A"B". 

In  this  projection  we  notice  that  no  part  of  the  pro- 
jected segment  lies  between  A"  and  B"  in  the  finite,  but 
that  the  projection  of  the  line-segment  AB  extends  from 
A"  upwards  to  C"  at  infinity,  and  thence  returns,  from 
below,  to  B". 

We  have  thus  reversed  the  segments  of  the  original 
line,  so  that  the  finite  part  ACB  extends  through  infin- 


PERSPECTIVE  PROJECTION.         173 

ity,  and  the  infinite  part  BDA  becomes  finite ;  and  thus 
D"  represents  the  external  point  of  bisection  of  AB,  that 
is,  the  point  at  infinity,  projected  into  the  finite.  We 
have  thus  a  graphic  illustration  of  the  theorem  that  the 
point  at  infinity  on  any  line-segment  is  the  external 
point  of  bisection  of  the  segment. 

Cor.  1.  A  circle  may  be  so  projected  that  any  point 
within  it  may  become  the  centre  of  the  curve  of  pro- 
jection. 

Let  C  be  the  given  point  in  the  circle,  and  let  ACB 
be  a  diameter,  and  ECF  a  chord  perpendicular  to  this 
diameter. 

On  any  plane  parallel  to  ECF  project  the  segment 
ACB  so  that  C  may  be  the  centre  of  A'B',  by  1. 

In  this  case  no  part  of  the  circle  goes  to  infinity,  and 
the  projection  is  a  closed  curve. 

Cor.  2.  A  circle  may  be  so  projected  that  any  point 
without  it  may  become  the  centre  of  the  figure  of  pro- 
jection. 

Apply  the  principle  involved  in  2. 

In  this  case  the  circle  becomes  two  curves  which 
extend  to  infinity  in  opposite  directions,  and  which  do 
not  meet  one  another.  The  diameter  AB  projects  into 
a  common  axis  to  the  two  curves,  and  the  projection  of 
the  given  point  bisects  the  part  of  the  common  axis 
intercepted  between  the  curves. 

EXERCISES  N. 

1.  Show  that  a  tetragram  may  be  projected  into  a  rectangle. 

2.  A  given  line  may  be  projected  to  infinity,  and  two  given 
angles  be  projected  into  required  angles. 


174  SOLID   OR   SPATIAL  GEOMETRY. 

3.  ABC  is  a  triangle,  and  DE  is  parallel  to  AG^  D  being  on 
BC,  and  E  on  AB.  CE  and  AD  intersect  in  0.  Then  BO  is 
a  median;  and  if  50  meets  DE  in  P  and  AC  in  Q,  BPOQ  is 
a  harmonic  range. 

4.  ABC  is  a  triangle,  and  AD,  BE  and  CF  are  parallel,  D 
being  on  ^BC,  i^on  AB,  and  £  on  AC.    Show  that 

^^ :  ^C  =  ^F- 5D  :  5i^.  CZ). 

5.  A  chord  of  a  circle  is  projected  to  infinity.  Show  that  the 
pole  of  the  chord  becomes  the  point  of  intersection  of  tangents 
which  touch  the  projection  at  infinity. 


SECTION   2. 
Plane  Sections. 

188.  The  definition  of  a  plane  section,  and  some 
general  facts  with  regard  to  it,  are  given  in  an  earlier 
portion  of  this  work  (see  Arts.  19  et  seq.). 

Evidently  the  plane  section  of  any  polyhedron  is  a 
polygon,  and  the  plane  section  of  a  sphere  is  a  circle. 
Such  sections  offer  no  distinctive  features  other  than 
what  belong  in  general  to  polygons  and  circles. 

But  when  we  make  plane  sections  of  the  cone  or 
cylinder,  we  are  introduced  to  curved  figures  which  are 
not  circles,  and  with  which  we  have  not  hitherto  become 
acquainted. 

These  we  propose  to  consider. 

Plane  Sections  of  the  Circular  Cone. 

189.  The  spatial  projection  of  a  circle,  from  a  centre 
of  projection  for  which  the  circle  is  a  cone-circle,  is  a  cir- 
cular cone.  The  section  of  this  cone  by  a  plane,  variable 
in  direction,  is  a  variable  curve,  which,  in  passing  through 
several  distinctive  phases  in  its  variation,  constitutes  a 
class  of  plane  curves  which  are  known  as  co7iic  sections, 
or  simply  conies. 

Hence  a  circle  may  be  projected  into  any  conic ;  and 
conversely,  any  conic  can  be  projected  into  a  circle. 
And  thus  any  conic  may  be  projected  into  any  other 
conic. 

175 


176 


SOLID   OR   SPATIAL   GEOMETRY. 


Classification  of  Conics. 

190.  Let  0  be  the  centre  of  the  circular  cone  0  •  AKB, 
L  being  a  generating  line,  and  let  V  denote  a  plane  of 
section  passing  through  any 
point  Q.     Then, 

1.  When  V  is  normal  to 
the  axis  of  the  cone,  the  sec- 
tion is  a  circle,  DCQ,  or  C. 

Now,  let  a  tangent  to  the 
circle,  C,  at  Q  be  drawn  in 
the  plane  V,  and  let  the 
plane  revolve  about  this  tan- 
gent line  as  an  axis.     Then, 

2.  When  V  makes  with 
the  axis  of  the  cone  an  angle 
less  than  a  right  angle,  and 
greater  than  the  semivertical 
angle  of  the  cone,  the  section 
is  an  Ellipse,  E. 

In  this  case  the  section- 
plane,  F,  cuts  completely- 
through  one  nappe  of  the  cone,  and  does  not  meet  the 
other  nappe.  Hence  the  ellipse  consists  of  a  single 
closed  curve,  as  represented  by  the  figure  E. 

3.  When  V  makes  with  the  axis  of  the  cone  an  angle 
equal  to  the  semivertical  angle  of  the  cone,  the  section  is 
a  Parabola,  P. 

In  this  case  V  is  parallel  to  a  single  generating  line, 
and  cuts  only  one  nappe  of  the  cone,  but  does  not  cut 
through  it,  and  thus  the  curve  extends  indefinitely  out- 
wards in  one  direction. 


CLASSIFICATION   OF   CONICS.  177 

The  parabola  is  consequently  a  single  curve,  but  not  a 
closed  curve. 

4.  When  V  makes  with  the  axis  an  angle  less  than  the 
semivertical  angle  of  the  cone,  the  section  is  a  Hyperbola. 

In  this  case  V  is  parallel  to  two  generating  lines 
which  make  a  finite  angle  with  one  another,  and  cuts 
into  both  nappes  of  the  cone,  but  does  not  cut  through 
either. 

Therefore,  the  hyperbola,  H,  consists  of  two  curves,  or 
rather  two  branches  extending  infinitely  outwards  in 
opposite  directions,  and  separated  from  one  another  by  a 
finite  interspace,  QQ'. 

191.  Degraded  forms.  All  the  conies  may  take  what 
are  called  degraded  forms ;  that  is,  forms  which  they 
assume  as  limiting  forms,  under  the  sequence  of  varia- 
tion, but  which  are  not  visible  curves. 

1.  Suppose  that  while  the  different  directions  of  the 
section  plane,  which  give  the  several  conies,  remain  the 
same,  the  section  plane  moves  up  to  0. 

Then,  (a)  The  circle  and  ellipse  reduce  to  a  point 
called  a  point-circle  and  a  point-ellipse  respectively.  Of 
course  there  is  no  final  distinction  between  a  point-circle 
and  a  point-ellipse,  but  their  names  indicate  their  origin. 

(6)  The  plane  of  the  parabola  becomes  a  tangent 
plane  to  the  cone,  and  touches  the  cone  along  two  coinci- 
dent lines ;  and  thus  the  parabola  degrades  into  two 
coincident  lines. 

(c)  The  plane  of  the  hyperbola  gives  in  section  two 
generating  lines  which  make  a  finite   angle  with  one 


178        SOLID  OR  SPATIAL  GEOMETRY. 

another,  and  the  hyperbola  thus  degrades  into  a  pair  of 
intersecting  lines. 

Hence  a  pair  of  intersecting  lines  is  frequently  called 
a  rectilinear  hyperbola. 

Let  V  be  the  plane  which  gives  the  hyperbola  H,  H 
(Fig.  190),  and  let  V  be  parallel  to  Fand  pass  through  0. 
V  gives  in  section  the  rectilinear  hyperbola  which  cor- 
responds to  //,  H\  and  if  we  draw  OT  to  the  middle 
point  of  QQ',  and  by  parallel  projection  in  the  direction 
TO,  we  project  the  hyperbola  H,  H  upon  the  plane  V, 
we  have,  on  that  plane,  a  hyperbola  and  its  correspond- 
ing rectilinear  hyperbola.  The  two  lines  which  form 
the  latter  are  then  called  the  asymptotes  of  the  former. 

2.  If  G  remains  fixed  while  Q  moves  up  to  0,  the 
ellipse  becomes  a  double  line-segment,  and  is  called  a 
line-ellipse. 

192.  From  the  generation  of  the  various  conies,  as 
now  explained,  we  deduce  — 

1.  That  the  circle  is  a  special  form  of  the  ellipse,  and 
that  properties  of  the  circle  are  special  cases  of  more 
general  properties  belonging  to  the  ellipse.  And  as  only 
one  direction  of  the  plane  of  section,  relatively  to  the 
axis  of  the  cone,  can  give  the  circle,  the  circle  has  only 
one  form,  or  all  circles  are  similar  to  one  another. 

2.  That  the  parabola  stands  intermediate  between"  the 
ellipse  and  the  hyperbola,  and  is  the  form  through  which 
one  of  these  curves  passes  into  the  other.  Also,  since 
only  one  direction  of  the  plane  of  section  relatively  to 
the  axis  of  the  cone  can  give  the  parabola,  the  curve  has 
only  one  form,  and  all  parabolas  are  similar  to  one  another. 


COMMON   PROPERTIES   OF   CONICS.  179 

3.  That  both  the  ellipse  and  the  hyperbola  are  varia- 
ble in  form,  the  ellipse  varying  from  the  circle  at  the 
one  limit  to  the  parabola  at  the  other,  and  the  hyperbola 
varying  from  the  parabola  at  the  one  limit  to  the  form 
of  two  coincident  lines  at  the  other. 

4.  That  all  the  conies  have  many  properties  in  common. 

Common  Properties  of  Conics. 

193.  As  a  line  can  meet  a  circular  cone  but  twice  (72), 
so  a  line  can  meet  a  conic  section  in  two  and  only  two 
points.  When  these  two  points  become  coincident,  the 
line  becomes  a  tangent  line,  and  the  point  of  contact  is 
a  double  point. 

The  conics  constitute  a  distinct  class  of  curves.  Being 
the  simplest  curves  that  it  is  possible  to  have,  they  are 
called  curves  of  the  first  order. 

All  curves  not  conics  belong  to  a  higher  order,  and 
cannot  be  obtained  as  sections  of  a  circular  cone  by  a 
plane,  nor  as  sections  of  any  cone,  one  of  whose  sections 
is  a  conic. 

Curves  are  classified  according  to  the  number  of  times 
they  may  be  met  by  a  line  under  the  most  favourable  cir- 
cumstances, and  all  curves  other  than  conics  can  be  met 
by  a  line  in  more  than  two  points,  either  real  or  imag- 
inary. 

Cor.  A  tangent  to  a  conic  lies  completely  without  the 
conic,  except  at  the  point  of  contact. 

194.  Z  is  a  sphere,  and  0  •  PFO  is  a  tangent  cone 
touching  the  sphere  in  the  small  circle  BEC. 


180 


SOLID   OR   SPATIAL  GEOMETRY. 


Denote  the  plane  of  BEC  by  U,  and  let  F  be  a 
plane  of  section  of  the  cone,  touching  the  sphere  in  S, 
and  meeting  U  in  the 
line  DH. 

Let  TTdenote  the  plane 
containing  the  axis  of  the 
cone  and  being  normal 
to  DH.  Then  W  is  per- 
pendicular to  both  U 
and  V. 

AQP  is  the  conic 
formed  by  F,  and  P  is 
any  point  on  this  conic. 
BD  is  a  centre  line  of 
the  circle  BEC,  and  is 
the  common  line  of  U 
and  W,  and  SA  is  the  common  line  of  W  and  F 

This  latter  line,  SA,  is  the  principal  axis  of  the  conic, 
or  simply  the  axis,  and  it  is  perpendicular  to  DH. 

The  conic  is  evidently  symmetrical  about  the  axis  SA, 
or  the  principal  axis. 

Draw  P^±  to  DH  and  Pilif  parallel  to  DH. 

Then,  P,  M,  D,  H  are  complanar,  all  lying  in  V,  and 
MH  is  a  rectangle,  and  therefore  PH=  MD. 

Join  PS  and  PO,  and  pass  a  plane  through  P,  par- 
allel to  U,  giving  the  circular  section  GPF,  and  meeting 
TF  along  the  line  GF. 

Then  BEC  and  GPF  being  cone-circles  with  0  as 
vertex, 

0P=  OF,  and  OE=OC; 


CF=EP=  SP. 


(Art.  86.  1.) 


COMMON  PEOPERTIES   OF   CONICS.  181 

Similarly,  SA  =  CA. 

But,  from  similar  triangles  CAD  and  FAM, 
CF :  MD  =  CA  :  AD, 
or  jSP:PH  =  SA:AD. 

But  the  ratio  ^S*^  :  AD  is  independent  of  the  position 
of  P  on  the  conic,  and  remains  constant  while  P  moves 
along  the  conic. 

Therefore,  denoting  this  constant  by  e,  we  have 

op 

=  e  =  a  constant. 

PH 

Hence  a  conic,  considered  as  the  locus  of  a  variable 
point,  may  be  delined  as  follows  : 

A  conic  is  the  locus  of  a  point  which,  being  confined 
to  one  plane,  so  moves  that  its  distance  from  a  fixed 
point  (S)  is  in  a  constant  ratio  (e)  to  its  distance  from 
a  fixed  line  (DH),  all  being  complanar. 

This  definition  is  usually  adopted  in  analytical  conies, 
and  it  is  sufficiently  general  to  include  every  conic. 

Def.  The  point  S  is  the  focus,  and  the  line  DH  is  the 
directrix.  A  is  the  vertex  of  the  conic,  and  the  constant 
e  is  the  eccentricity. 

PM  is  an  ordinate  to  the  principal  axis,  and  PS  is  the 
focal  distance  of  the  point  P. 

195.  Let  the  accompanying  figure  represent  the  sec- 
tion by  the  plane  W. 

A  second  sphere,  Z',  may  be  drawn,  to  touch  the  cone 
and  the  plane  V  at  SK  Then  BC  and  B'O  representing 
the  sections  of  the  circles  of  contact,  the  planes  of  these 


182 


SOLID   OR   SPATIAL   GEOMETRY. 


circles  are  parallel,  and  they  cut  the  plane  V  in  parallel 
lines  represented  in  section  at  D  and  D'.  Hence  the 
conic  has  two  foci,  S  and  S',  two  vertices,  A  and  A',  and 
two  parallel  directrices  represented  in  section  by  D 
and  D'. 

The  figure  as  here  drawn  applies  particularly  to  the 
ellipse,  but  it  may  serve  as  a  type  for  all  the  other  conies. 


In  the  parabola  one  vertex,  focus,  and  directrix  are  at 
infinity. 

In  the  hyperbola  the  second  focus  is  given  by  the 
point  of  contact  of  the  sphere  Z". 

Thus  in  the  ellipse  the  curve  lies  between  the  direc- 
trices, while  in  the  hyperbola  the  directrices  lie  between 
the  vertices  of  the  two  branches  of  the  curve. 


COMMON   PROPERTIES   OF   CONICS.  183 

196.  In  the  figure  of  Art.  195,  considered  as  a  plane 
figure,  Z  is  the  incircle,  and  Z  is  an  excircle  of  the 
triangle  AOA!.  Therefore,  AS=A}S'  (P.  Art.  135. 
Ex.  1)  ;  so  that  both  foci  are  similarly  situated  with 
respect  to  the  corresponding  vertices. 

Also,  drawing  A!T  parallel  to  QA, 

AS'  =  A'B'  =  A'T=AS  =  AG. 

Therefore,  the  triangles  D'A'T  and  DAG  are  con- 
gruent, and  A'D'  =  AD ;  or  the  directrices  are  similarly- 
situated  with  respect  to  the  corresponding  foci  and 
vertices. 

Hence  it  follows  that  the  same  curve  may  be  drawn 
from  either  vertex  with  the  corresponding  focus  and 
directrix,  and  therefore  that  a  line  drawn  at  right  angles 
to  the  principal  axis,  and  bisecting  the  distance  between 
the  foci,  is  an  axis  of  symmetry  of  the  curve. 

And  as  the  principal  axis  is  also  an  axis  of  symmetry, 
a  conic  has,  in  general,  two  axes  of  symmetry  bisecting 
one  another  at  right  angles.  These  are  called  the  axes 
of  the  conic. 

In  the  parabola  one  axis  is  at  infinity. 

197.  The  character  of  the  conic  is  determined  by  the 
value  of  its  eccentricity. 

In  the  figure  to  Art.  194, 

e  =  SA:AD=CA:  AD. 

1.  Let  AD  be  infinite.      Then  e  =  0,  and  the  plane  V 
is  parallel  to  the  plane  U,  and  the  conic  is  a  circle. 
Therefore,  the  eccentricity  of  a  circle  is  zero. 
In  this  case  both  spheres  touch  V  at  the  centre  of  the 


184  SOLID   OR   SPATIAL   GEOMETRY. 

circular  section,  and  the  foci  of  the  circle  become  comci- 
dent  at  the  centre  of  the  circle. 

2.  Let  AD=CA.  Then  e  =  l;  and  the  triangle  CAD 
being  isosceles, 

ZADC=ZACD=ZBCO  =  ZOBa 

Therefore,  AD  is  parallel  to  OB,  and  the  conic  is  a 
parabola. 

Hence  the  eccentricity  of  the  parabola  is  unity.  In 
this  case  a  second  sphere  cannot  be  drawn  in  any  finite 
position  so  as  to  satisfy  the  conditions  for  a  focus. 

3.  When  AD  is  <  oo  and  >  CA,  the  value  of  e  lies 
between  zero  and  unity.  The  ZACD  is>  the  ZADC, 
and  the  plane  V,  being  inclined  to  the  axis  of  the  cone 
at  an  angle  greater  than  the  semivertical  angle  of  the 
cone,  cuts  through  one  nappe  and  gives  the  ellipse. 

4.  When  AD  is  <AC  and>0,  e  lies  between  unity 
and  infinity  ;  the  angle  ADC  is  greater  than  ACD,  and 
the  plane  V,  being  inclined  to  the  axis  of  the  cone  at  an 
angle  less  than  the  semivertical  angle  of  the  cone,  cuts 
into  both  nappes  and  gives  the  hyperbola. 

198.  When  the  centre  of  a  circular  cone  goes  to 
infinity,  the  cone  becomes  a  cylinder,  and  the  only  possi- 
ble plane  sections  are  the 
circle,  the  ellipse,  and  two 
parallel  lines  representing 
the  parabola  and  hyperbola. 

The  ellipse,  including  the 
circle  as  a  particular  case, 
is  the  most  important  of 
the  conic  sections,  and  we  propose  to  develop  some  of  its 


THE  ELLIPSE.  185 

more  prominent  properties,  through  its  relationship  to 
the  circular  cylinder. 

Let  ADB  be  one-half  of  the  right  section  of  a  circular 
cylinder,  and  let  AEB  be  the  corresponding  half  of  an 
oblique  section  of  the  same  cylinder.  Then  ADB  is  a 
semicircle  with  AB  as  diameter,  and  AJEB  is  one-half 
of  an  ellipse. 

From  C,  the  centre  of  both  the  circle  and  the  ellipse, 
draw  CD  and  CE  perpendicular  to  AB,  the  former  in  the 
plane  of  the  circle,  and  the  latter  in  that  of  the  ellipse. 
Then  ED  is  a  segment  of  a  generating  line  of  the  cylinder. 

On  the  ellipse  take  any  point  P,  and  draw  PQ  parallel 
to  ED,  and  QG  parallel  to  DC. 

The  A  ECD  and  FGQ  are  similar,  and  CD=CQ  = 
the  radius  of  the  cylinder,  and  CD  is  >  GQ. 

Therefore,  ED  is  >  PQ. 

But  CE'=CD^+DE'=CQ'-\-DE',  and  this  is  >Cg'+ 
PQ\ 

Whence  CE  is  >  CP. 

Or,  CE  is  the  longest  segment  from  C  to  the  ellipse, 
and  is  the  semiaxis-major  of  the  ellipse. 

In  like  manner  it  may  be  shown  that  CP  is  >  CA; 
or  that  CA  is  the  shortest  segment  from  the  centre  to 
the  ellipse,  and  is  the  semiaxis-minor  of  the  ellipse. 

These  axes  are  perpendicular  to  one  another. 

199.  On  account  of  the  similar  triangles,  PGQ  and 
EDC  (Fig.  of  198), 

PG.GQ=EC:CD. 
But  EC :  CD  is  constant  for  a  constant  direction  of  the 
plane  of  oblique  section. 

.-.  PG :  GQ  =  a.  constant. 


186 


SOLID   Oil   SPATIAL   GEOMETRY. 


=  -— -  =  constant. 


Hence  the  following  construction  for  an  ellipse  ;  AQD 
is  a  quadrant  of  a  circle  with  cen- 
tre C,  and  GQ  is  a  chord  A.  to  AC. 
Take  GP,  a  constant  multiple  of 
GQ.  The  locus  of  P  is  an  ellipse 
whose  semiaxis-minor  is  AC. 

Again,  draw  PH  A.  to  CE  and 
let  CQ  meet  HP  in  P. 

Then,    from    similar    triangles 
EPQ  and  CGQ, 

RP^PQ. 
GC     GQ' 

RH^PG 

"  PH     QG 

And  PH  is  a  constant  part  of  PH;  hence  the  the- 
orem— 

If  on  a  chord  of  a  circle,  perpendicular  to  a  fixed 
diameter,  a  point  be  taken  so  as  to  divide  the  chord  in  a 
constant  ratio,  the  locus  of  the  point  is  an  ellipse,  and 
the  fixed  diameter  is  the  major  or  the  minor  axis  of  the 
ellipse,  according  as  the  point  divides  the  chord  inter- 
nally or  externally". 

Def.  The  circles  which  have  the  major  and  minor  axes 
of  the  ellipse  as  their  diameters  are  the  major  and  minor 
auxiliary  circles  to  the  ellipse. 

200.  In  the  figure  of  Art.  198,  let  AEB  be  a  semicircle 
with  AB  as  diameter  and  C  as  centre,  and  let  CE  be 
perpendicular  to  AB. 

Project  orthogonally  on  any  plane,  V,  passing  through 
AB.     Then  GQ=  GP  cos  PGQ,  and  as  PGQ  is  sl  con- 


THE   ELLIPSE. 


187 


stant  angle,  G^Q  is  in  a  constant  ratio  to  GP  and  is  less 
than  GP. 

Hence  Q  lies  on  an  ellipse  of  which  AB  is  the  major 
axis. 

Therefore,  the  orthogonal  projection  of  the  circle  on 
any  plane  not  parallel  to  its  own  is  an  ellipse. 

This  result  also  follows  directly  from  the  statements 
of  Art.  190,  since,  in  general,  the  perspective  projection 
of  a  circle  on  any  plane  which  cuts  through  one  nappe  is 
an  ellipse.  But  in  the  case  of  perspective  projection  the 
same  diameter  of  the  circle  does  not  project  into  an  axis 
of  the  ellipse, 

201.  Conjugate  diameters.  Let  ADBE  be  a  right 
section  of  a  circular  cylinder,  by  the  plane  U,  and  let 
adbe  be  an  oblique  section  by  the  plane  V.  Also  let  AB 
and  DE  be  perpendicular  diam- 
eters of  the  circle,  and  FG  be  a 
chord  parallel  to  AB. 

Now  let  the  whole  figure  on 
Uhe,  projected  ant-orthogonally 
(Art.  177)  on  V. 

Then  c  is  the  centre  of  the 
ellipse  and  ah  and  de,  the  pro- 
jections of  AB  and  DE,  are  a 
pair  of  conjugate  diameters  of 
the  ellipse, 

And  FG  is  bisected  at  H.  Whence,  from  the  nature 
of  parallel  projection,  fg  is  bisected  at  h,  and  is  parallel 
to  ah. 

Therefore,  de  bisects  all  chords  parallel  to  ah  ;  and  in 
like  manner  ah  bisects  all  chords  parallel  to  de. 


188  SOLID   OR   SPATIAL   GEOMETRY. 

Hence,  when  two  diameters  are  conjugate,  each  bisects 
all  chords  parallel  to  the  other. 

Cor.  1.  Conjugate  diameters  are  such  that  each  is 
parallel  to  the  tangents  at  the  extremities  of  the  other. 

Cor.  2.  Conjugate  diameters  in  the  ellipse  are  the 
parallel  projections  of  orthogonal  diameters  in  that  circle 
whose  projection  gives  the  ellipse. 

Manifestly  an  indefinite  number  of  pairs  of  conjugate 
diameters  may  be  found,  and  each  diameter  has  one,  and 
only  one,  conjugate. 

Cor.  3.  When  the  plane  V  is  parallel  to  AB  or  DE, 
ah  and  de  are  perpendicular  to  one  another,  and  become 
the  principal  diameters  of  the  ellipse. 

202.  Since  Aa,  Bh,  etc.,  are  generating  lines  of  the 
cylinder,  they  are  lines  of  contact  of  tangent  planes 
(Art.  73).  Let  four  tangent  planes  to  the  cylinder  touch 
it  at  Aa,  Bh,  Dd,  and  Ee.  The  section  of  these  by  CJis 
a  square,  whose  sides  are  parallel  to  AB  and  DE;  and 
this  section  remains  constant  in  area  however  AB  and 
DE  may  be  drawn,  provided  they  are  diameters  which 
intersect  orthogonally. 

The  section  of  the  tangent  planes  by  F"  is  a  parallelo- 
gram whose  sides  are  parallel  to  ah  and  de. 

Now  (Art.  116),  the  area  of  the  square  is  equal  to  that 
of  the  parallelogram  multiplied  by  the  cosine  of  the 
angle  between  U  and  V. 

Therefore,  the  area  of  the  parallelogram  is  unaffected 
by  any  change  in  the  position  of  V,  which  does  not 
change  the  inclination  of  that  plane  to  the  axis  of  the 


THE   ELLIPSE.  189 

cylinder ;  that  is,  whicli  does  not  change  the  form  of  the 
ellipse. 

Hence,  in  any  given  ellipse  the  parallelogram  formed 
by  tangents  at  the  extremities  of  conjugate  diameters, 
or  the  parallelogram  on  a  pair  of  conjugate  diameters 
taken  in  both  length  and  direction,  is  constant. 

Cor.  If  a  and  h  denote  the  semiaxes,  and  a'  and  6' 
denote  a  pair  of  conjugate  semidiameters,  and  6  be  the 
angle  between  them, 

a'h'  sva.6=db. 

203.  Let  APQB  be  a  semicircle  on  a  plane,  U,  and 
let  CP  and  CQ  be  radii  at  right  angles  to  one  another. 
Let  the  whole  be  projected 
orthogonally  upon  a  plane,  V, 
passing  through  AB,  and  in- 
clined to  Vat  a  fixed  angle,  $. 
And  let  CP',  CQ'  be  the  pro- 
jections of  CP  and  CQ  respec- 
tively. 

Then  (Art.  201.  Cor.  2),  AP'Q'B  is  a  semiellipse,  and 
CP'  and  CQ'  are  a  pair  of  semi-conjugate  diameters. 

Draw  PE  and  QF 1.  to  AB,  and  join  P'E  and  Q'F. 
Then  P'E  and  Q'F  are  ±  to  ^B ;  and  since  PCQ  is  a  right 
angle,  CF=  EP,  and  EC  =  QF.  Also,  PP'  =  EP  sin  d, 
and  QQ'  =  FQ  sin  e=CE  sin  6. 

And  CP'^  +  CQ"  =  CE^  -f-  EP^  -  PP'^ 

+  CF^  +  F(^  -  QQ'^ 
=  2CP'-  (EP'  -f-  FQ"")  sin^  $ 
=  CP- (2  -  shi' 6) 
=  a  constant. 


190  SOLID   OR   SPATIAL   GEOMETRY. 

Therefore,  the  sum  of  the  squares  on  a  pair  of  con- 
jugate diameters  of  an  ellipse  is  constant. 

Cor.     Denoting  the  parts  as  in  Art.  202.  Cor., 
a'2  +  6'2  =  a-  +  h\ 
The  results   of  Arts.  202  and  203  are  known  as  the 
theorems  of  Apollonius. 

204.  Let  F  be  a  plane,  cutting  a  circular  cone  so  as 
to  give  an  ellipse,  and  let  P  be  any  point  on  the  ellipse. 
P  is  a  point  on  the  cone. 

Let  Z  be  the  sphere  which  touches  Fat  the  focus  S, 
and  Z'  be  the  sphere  which  touches  Fat  the  focus  S';  also 
let  K  denote  the  circle  of  contact  of  Z  with  the  cone, 
and  K^  denote  the  circle  of  contact  of  Z'. 

Draw  through  P  a  generating  line  of  the  cone.  This 
line  cuts  Kin  k,  and  K'  in  k',  and  at  these  points  touches 
the  spheres  Z  and  Z'. 

But  K  and  K'  being  cone  circles  to  the  vertex  0,  kk' 
is  constant  for  all  positions  of  the  generating  line. 

But  Pk  =  PS  and  Pk'  =  PS',  being  tangents  to  the 
spheres.  .^  ^p^  p^,  _  ^  constant. 

Therefore,  in  any  given  ellipse,  the  sum  of  the  dis- 
tances of  any  point  on  the  curve  from  the  two  foci  is 
constant. 

205.  The  result  of  the  preceding  article  furnishes  a 
convenient  practical  method  of  drawing  an  ellipse. 

Over  two  pins  placed  at  S  and  S'  put  a  loop  of  inex- 
tensible  thread,  and  keep  it  stretched  by  a  pencil  at  P. 
The  locus  of  P  is  an  ellipse  of  which  S  and  S'  are 
the  foci. 


THE  ELLIPSE.  191 

For  the  whole  length  of  thread  being  constant,  and 
the  part  SS'  being  constant,  it  follows  that  SP  +  PS'  is 
constant. 

Cor.  By  considering 
the  phase  when  P  comes 
to  A  or  to  B,  we  readily 

see  that  SB-\-BS'=AB,    a~s  c s^ 

and  hence  that  the  whole  length  of  thread  is  2  AS',  or 
2BS. 

206.  Let  PT  be  a  tangent  to  the  ellipse  at  P,  and  let 
Q  be  any  point,  other  than  P,  on  this  tangent.  Then, 
since  the  tangent  has  only  one  point  in  common  with 
the  curve  (Art.  193.  Cor.),  QS'  cuts  the  curve  in  some 
point  P. 

Then,  SQ  +  QS'  is  >  SB  +  RS> ; 

or  SQ  +  QS'  is  >  SP  +  PS'. 

Hence  SP-\-PS'  is  the  shortest  route  from  S  to  S'  by 
way  of  the  line  PT,  and  hence  SP  and  S'P  are  equally 
inclined  to  PT. 

Therefore,  in  any  ellipse,  the  lines  from  the  foci  to 
the  point  of  contact  of  any  tangent  are  equally  inclined 
to  the  tangent. 

207.  0  •  GQH  is  a  circular  cone ;  APQB  is  an  elliptic 
section,  and  CPD  and  EQF  are  circular  sections  cutting 
the  elliptic  section  in  the  lines  PM  and  QN. 

CD  and  EF  are  diameters  of  the  circles,  and  AB  is  the 
axis  of  the  ellipse,  all  lying  in  the  plane  W  (Art.  194). 
PM  and  QN  are  perpendicular  to  AB  and  to  the  lines 
CD  and  EF  respectively. 


192 


SOLID   OR   SPATIAL   GEOMETRY. 


Therefore         CM-  MD  =  PM^,  and  EN-  NF=  QN- 

But    A  AMD  -  A  ANF,  and  A  ENB  ^  A  CMB. 
.:  AM:MD  =  AN.NF; 
and  MB:GM=NB:  EN. 

Therefore,  by  multiplication, 

AM-  MB  :  MD  ■  CM=  AN-  NB-.NF-  EN; 
or  AM-  MB :  PM^  =  AN-  NB :  QN^. 

In  a  similar  manner  it 
is  proved  that  a  like  rela- 
tion holds  true  for  the 
hyperbola.     Therefore, 

In  the  ellipse  and  the 
hyperbola,  if  perpendicu- 
lars be  drawn  from  points 
on  the  figure  to  the  prin- 
cipal axis,  the  squares 
on  these  perpendiculars 
are  proportional  to  the 
rectangles  on  the  parts 
into  which  each  perpen- 
dicular divides  the  prin- 
cipal axis. 

Again,  let  A'P'N'  be  a  parabolic  section 
is  parallel  to  OG,  and  EM'  =  GN'. 

But  from  the  similar  A  A'M'F  and  A'N'H, 
A'M'  .M'F=  A'N' :  N'H. 

And  1 :  EM'  =  1 :  GN'. 

.-.  A'M' :  EM'  -  M'F=  A'N' :  GN'  •  N'H-, 
or  A'M 'iP'M'^^  A'N' :  Q'N". 


Then  A'N' 


THE  PARABOLA.  193 

Hence,  in  the  parabola,  the  squares  on  perpendiculars 
from  points  on  the  figure  to  the  axis  are  proportional  to 
the  parts  of  the  axis  intercepted  between  the  vertex  of 
the  parabola  and  the  foot  of  each  perpendicular. 

Def.  The  perpendiculars  of  this  article  are  called 
ordinates  to  the  principal  axis,  and  the  segments  into 
which  they  divide  the  axis  are  called  abscissoe.  So  that 
in  the  ellipse  and  hyperbola  the  square  on  any  ordinate 
is  in  a  constant  ratio  to  the  rectangle  on  its  abscissae. 

In  the  parabola,  however,  one  abscissa  is  infinite,  and 
we  have  only  the  finite  one  to  consider.  Then,  the  square 
on  an  ordinate  is  in  a  constant  ratio  to  its  abscissa. 

EXERCISES   O. 

1.  Show  that  the  area  of  an  ellipse  is  wab. 

2.  A  right  cylinder  has  its  base  an  ellipse  with  axes  a  and  b. 
Show  how  to  cut  it  by  a  plane  so  that  the  section  may  be  a  circle. 

3.  If  P  be  a  point  on  a  hyperbola,  of  which  S  and  F  are  foci, 
then  SP  —  PF  =  constant. 

208.  As  the  parabola  is  a  limiting  form  of  the  ellipse 
(Art.  192.  2),  the  fundamental  properties  of  the  parabola 
may  be  obtained  from  those  of  the  ellipse  by  supposing 
that  one  focus  of  the  ellipse  goes  to  infinity,  while  the 
other  focus  remains  at  a  finite  distance  from  the  vertex. 

We  shall,  however,  obtain  these  relations  by  means  of 
the  perspective  projection  of  the  circle. 

In  the  accompanying  diagram,  the  right-hand  figure  is 
the  projection  of  the  left-hand  one,  and  for  the  sake  of 
convenience  in  comparison,  a  point  and  its  projection 
are  denoted  by  the  same  letter  in  both  figures. 


194 


SOLID   OR   SPATIAL   GEOMETRY. 


In  (I),  BB'  is  a  tangent  to  the  circle  APB,  touching 
it  at  B,  and  BAK  is  a  centre  line.  B'PQ  is  any  secant 
line  from  B',  and  PT  and  QT  ave  tangents  meeting  at  T. 

B'Vis  a  tangent  from  B'  touching  the  circle  at  V,  and 
TK  is  perpendicular  to  BA. 

The  circle  is  projected  into  a  circular  cone,  and  the 
plane  which  gives  the  parabola  in  section  touches  the 
circle  at  A  and  is  parallel  to  the  tangent  BB',  and  BB' 
is  projected  to  infinity. 


(I)  (II) 

In  the  projection  (II),  A  becomes  the  vertex  A  of  the 
parabola,  and  BB'  goes  to  infinity  (Art.  181.  Cor.  2),  and 
hence  the  lines  through  B  in  (I)  become  a  system  of 
parallels  in  the  projection  (II),  and  thus  in  (II)  KAGco 
is  the  axis  of  the  parabola,  and  TVHoc  is  parallel  to 
the  axis. 

So,  also,  the  lines  through  B'  in  (I)  project  into  a 
system  of  parallels  in   (II);   that  is,    QPB'  and    VB' 


COMMON   PROPERTIES   OF   CONICS.  195 

become  the  parallels  QPoo  and  Foo,  or  the  tangent  at 
Fis  parallel  to  PQ. 

Now  in  (I)  B'  is  pole  to  VB,  and  therefore  QHPB'  is 
a  harmonic  range  (P.  Art.  311.  2). 

Therefore,  also,  in  (II),  QHPcc  is  a  harmonic  range, 
and  H  is  the  middle  point  of  QP. 

Hence  1.  In  the  parabola,  any  line  parallel  to  the 
axis  is  a  diameter,  and  bisects  all  chords  parallel  to  the 
tangent  (Foo)  at  its  vertex.  Thus  PQ  is  bisected  by 
TVH. 

The  direction  of  the  diameter  conjugate  to  FH" oo  is 
given  by  PQ,  but  its  position  is  at  infinity. 

Again,  in  (I),  T  is  pole  to  PQ,  and  TVHB  is  a  har- 
monic range  (P.  Art.  311.  2). 

Therefore,  in  (II),  TVHao  is  harmonic,  and  F  bisects 
TH.     Hence: 

2.  In  the  parabola,  tangents  at  the  end-points  of  any 
chord  (PQ)  meet  upon  the  diameter  to  that  chord  (TH); 
and  the  part  of  the  diameter  intercepted  between  the 
chord  and  the  point  of  meeting  of  the  tangents  is  bisected 
by  the  curve .     TH  is  bisected  at  F 

Again,  in  (I),  TK\s  the  polar  of  G  (P.  Art.  267),  and 
hence  KAGB  is  a  harmonic  range. 

Therefore,  in  (II),  KAGco  is  harmonic,  and  A  bisects 
KG.     Hence : 

3.  In  the  parabola,  if  two  tangents  be  drawn  from 
any  point  to  the  curve,  and  a  perpendicular  be  drawn 
from  the  same  point  to  the  axis,  the  part  of  the  axis 
intercepted  between  the  foot  of  the  perpendicular  and 
the  chord  of  contact  {PQ)  is  bisected  by  the  vertex  of 
the  curve.    KG  is  bisected  at  A. 


196  SOLID   OR   SPATIAL   GEOMETRY. 

209.  Since  the  circle  can  be  projected  into  any  conic, 
and  anharmonic  properties  are  projected  without  change, 
all  the  properties  of  the  circle  which  depend  upon 
anharmonic  or  harmonic  relations  are  equally  true  for  all 
the  conies. 

Thus  the  theorems  in  plane  geometry,  given  in  Arts. 
311,  312,  313,  314,  and  many  of  the  following  ones,  are 
true  when  we  read  conic  for  circle. 

To  enter  any  further  into  this  subject  is  beyond  the 
scope  of  this  work.  The  student  who  desires  to  pursue 
this  most  interesting  subject  will  find  it  fully  developed 
in  Salmon's  'Conies,'  or  Cremona's  'Projective  Geometry,' 
or  in  Poncelet's  great  work,  the  'Traite  des  proprietes 
projective  des  figures.' 

EXERCISES  p. 

1.  In  the  parabola,  prove  that  the  tangent  at  any  point  on  the 
curve  makes  equal  angles  with  the  axis  and  the  line  joining  that 
point  to  the  focus. 

2.  P  being  a  point  on  a  parabola,  if  PM  be  drawn  perpendicu- 
lar to  the  axis  and  PiV  perpendicular  to  the  tangent  at  P,  the  part 
MN  intercepted  on  the  axis  is  constant. 

3.  The  tangent  at  the  vertex  of  a  parabola  bisects  the  part 
of  any  other  tangent  lying  between  the  point  of  contact  and  the 
axis. 

4.  The  tangent  at  the  vertex  of  a  parabola,  and  the  perpendicu- 
lar from  the  focus  upon  any  other  tangent,  meet  the  latter  tangent 
at  the  same  point. 

5.  In  the  projection  of  Art.  208,  if  TK  (1)  projects  into  the 
directrix,  then  G  will  project  into  the  focus. 


SECTION  3. 
Spheric  Geometry. 

210.  When  a  spatial  figure  is  cut  by  a  sphere,  the 
elements  common  to  the.  spatial  figure  and  the  sphere 
form  a  figure  which  lies  on  the  sphere  in  the  same 
manner  as  a  plane  figure  lies  upon  its  plane. 

Such  a  figure  is  a  spJieric  figure,  as  being  confined  to 
a  spherical  surface,  and  the  geometry  of  such  figures  is 
called  spheric  geometry  or  spherical  surface  geometry. 

On  account  of  the  uniform  curvature  of  a  sphere,  a 
spheric  figure  may  be  moved  about  upon  the  spherical 
surface  upon  which  it  lies,  without  undergoing  any  neces- 
sary change  in  the  relations  of  its  parts,  just  as  a  plane 
figure  may  be  moved  about  over  the  plane  surface  upon 
which  it  lies. 

There  are  many  analogies  between  spheric  geometry 
and  plane  geometry,  and  many  of  the  theorems  and  of 
the  methods  employed  are  more  or  less  alike.  But  there 
are  also  fundamental  differences. 

The  prominent  analogies  will  be  exhibited  in  the 
sequel. 

211.  It  must  be  understood  in  the  beginning  that 
spheric  geometry  does  not  deal  with  a  comparison  of 
the  properties  or  relations  of  figures  drawn  upon  differ- 
ent spheres ;  its  purpose  is  not  this,  but  to  investigate 

197 


198  SOLID   OR   SPATIAL   GEOMETRY. 

the  properties  which  belong  to  a  figure  in  consequence  of 
its  lying  upon  a  sphere. 

Hence  all  spheric  figures  are  supposed  to  lie  on  one 
and  the  same  sphere,  just  as  all  the  figures  in  plane 
geometry  are  supposed  to  lie  on  one  and  the  same  plane. 

The  radius  of  the  particular  sphere  is  altogether  arbi- 
trary, and,  except  in  the  case  of  metrical  theorems  or 
problems,  the  radius  may  be  left  out  of  the  consideration. 

The  centre  of  the  sphere  will  be  referred  to  as  the 
centre. 

212.  Every  section  of  a  sphere  by  a  plane  is  a  circle, 
and  when  the  plane  contains  the  centre  of  the  sphere, 
the  section  is  the  largest  circle  in  this  way  obtainable, 
and  is  called  a  great  circle  of  the  sphere. 

It  will  appear  hereafter  that  when  a  plane  figure 
involving  the  line  has  an  analogue  in  spheric  geometry, 
the  line  is  represented  by  the  great  circle.  And  as  there 
can  be  no  straight  line  in  connection  with  any  spheric 
figure,  we  shall,  for  the  sake  of  the  analogy,  commonly 
speak  of  a  great  circle  as  a  spheric  line.  Then  all  other 
circles  are  spheric  circles. 

Any  limited  part  of  a  spheric  line  is  a  spheric  arc,  and 
parts  of  other  circles  are  circular  arcs. 

213.  The  spheric  line,  unlike  a  planar  line,  returns 
into  itself  without  passing  to  infinity. 

Evidently  the  spheric  line  divides  the  whole  spherical 
surface  into  two  congruent  parts,  just  as  the  planar  line 
may  be  said  to  divide  the  whole  plane  into  two  congruent 
parts.  The  parts  of  the  plane,  however,  extend  to  infin- 
ity, while  those  of  the  spherical  surface  do  not. 


SPHERIC   GEOMETRY.  199 

We  have  here  a  fundamental  distinction  between  plane 
and  spheric  geometry,  namely,  that  spheric  geometry  has 
no  infinity. 

214.  As  any  plane  which  gives  in  section  a  spheric 
line  must  pass  through  the  centre,  any  two  points  on  the 
sphere,  not  in  line  with  the  centre,  determine,  with  the 
centre,  one  plane,  and  therefore  one  spheric  line. 

Thus,  like  a  planar  line,  a  spheric  line  is  determined 
by  any  two  points,  provided  they  are  not  collinear  with 
the  centre. 

And  through  any  two  points  not  collinear  with  the 
centre,  one,  and  only  one,  spheric  line  can  pass. 

When  two  points  are  in  line  with  the  centre,  the  three 
points  determine  only  one  line,  a  diameter  of  the  sphere, 
and  through  this  any  number  of  planes  can  pass,  giving 
in  section  the  same  number  of  spheric  lines. 

Now  in  plane  geometry,  any  two  points  determine  one 
line,  unless  the  points  be  at  infinity.  For  in  this  latter 
case,  since  all  parallels  of  a  system  meet  at  infinity,  two 
points  at  infinity,  in  opposite  directions,  determine  a 
system  of  parallels. 

We  see,  then,  that  for  two  points  to  be  collinear  with 
the  centre,  in  spheric  geometry,  is  analogous  to  two 
points  at  infinity  in  opposite  directions,  in  plane  geometry. 

Thus  spheric  lines  passing  through  a  pair  of  opposite 
points  are  analogous  to  parallel  lines  in  plane  geometry ; 
and  hence  there  is  no  theory  of  parallels  in  spheric  geometry 
as  in  plane  geometry. 

Cor.  Any  number  of  points  on  a  sphere,  no  two  of 
which  are  collinear  with  the  centre,  and  no  three  of 


200 


SOLID   OR   SPATIAL   GEOMETHY. 


which  are  complanar  with  the  centre,  determine  as  many- 
spheric  lines  as  there  are  groups  of  the  points  taken  two 
and  two. 

The  corresponding  theorem  in  plane  geometry  is,  that 
any  number  of  points,  no  two  of  which  are  at  infinity, 
and  no  three  of  which  are  in  line,  determine  as  many 
lines  as  there  are  groups  of  the  points  taken  two  and  two. 

215.  The  normal,  through  the  centre,  to  the  plane  of 
a  great  circle,  meets  the  sphere  in  two  opposite  points 
which  are  end-points  of  a  diam- 
eter. 

These  points  are  poles  of  the 
great  circle ;  and  in  relation  to 
the  poles  the  circle  is  called  the 
equator. 

Thus  every  spheric  line  has 
two  poles,  and  any  point  on  the 
sphere,  considered  as  a  pole, 
has  an  opposite  pole,  and  an 
equator. 

Thus  AB,  in  the  figure,  is  normal  to  the  plane  EGFH, 
and  passes  through  the  centre  0,  and  meets  the  sphere 
at  A  and  at  B.  Then  A  and  B  are  poles  of  the  spheric 
line  EGFH,  and  reciprocally  EGFH  is  the  equator  to 
the  poles  A  and  B. 

Evidently  the  angle  AOE,  subtended  at  the  centre 
between  a  pole  and  any  point  on  its  equator,  is  a  right 
angle,  and  the  spheric  arc  AE  is  one-fourth  of  a  whole 
circumference. 

If  a  quadrant  of  a  great  circle  has  one  extremity  fixed 
at  A  while  the  other  moves  over  the   surface  of  the 


SECTION   OF  TWO   PLANES.  201 

sphere,  the  moving  extremity  will  describe  the  great 
circle  or  spheric  line  EOFH,  which  is  the  equator  to  the 
point  A  as  pole. 

Hence  the  quadrant  AE  is  the  spheric  radius  of  the 
great  circle  described ;  or  the  spheric  radius  of  a  great 
circle  is  a  quadrant. 

Cor.  If  any  jjoint  P  be  taken  in  the  quadrant  AE, 
when  the  quadrant  moves  over  the  surface  having  A 
fixed,  P  will  describe  a  circle  PPQupon  the  sphere.  As 
the  arc  AP,  and  therefore  the  chord  AP,  is  constant,  PBQ 
is  a  cone-circle  to  A  as  vertex,  and  its  plane  is  normal  to 
AB,  and  therefore  parallel  to  the  plane  of  EOFH. 

And  as  A  and  B  are  points  on  the  sphere  from  which 
the  spheric  circle  PRQ  can  be  described  by  means  of 
constant  arcs,  AP  or  BP,  these  points  are  poles  to  the 
circle  PRQ,  and  the  arc  AP,  as  also  BP,  is  its  spheric 
radius. 

Thus  every  circle  on  a  sphere  has  two  spheric  radii 
which  are  supplementary  to  one  another,  and  the  poles 
of  any  great  circle  are  poles  to  all  spheric  circles  whose 
planes  are  parallel  to  that  of  the  great  circle. 

Section  of  Two  Planes. 

216.  The  sphere  which  has  its  centre  on  the  common 
line  of  two  intersecting  planes  has,  in  section  by  these 
planes,  two  spheric  lines  which  intersect  in  opposite 
points,  or  at  the  end-points  of  a  diameter. 

Thus  the  two  planes  which  have  in  common  the  line 
AB   (Fig.  of  215)   give  by  their  intersection  with  the 


202  SOLID   OR   SPATIAL   GEOMETRY. 

sphere  the  two  spheric  lines  AEBF  and  AGBH,  inter- 
secting in  A  and  B,  and  mutually  bisecting  one  another. 

The  angle  between  these  spheric  lines  is  equal  in 
measure  to  the  dihedral  angle  between  the  planes  which 
give  rise  to  the  lines.  But  if  EGFH  be  the  equator  to 
A,  EO  and  GO  are  each  perpendicular  to  AB,  and  there- 
fore EOG  measures  the  dihedral  angle  between  the 
planes,  and  hence  also  the  angle  between  the  spheric 
lines. 

But  the  angle  EOG  =  (aiVG  EG)-^  EO;  and  since  EO 
is  supposed  to  be  constant,  our  investigations  being  con- 
fined to  one  and  the  same  sphere, 

Therefore,  the  angle  between  two  spheric  lines  is 
proportional  to  the  arc  which  they  intercept  upon  the 
equator  to  their  points  of  intersection  as  poles. 

Cor.  1.  If,  through  A,  tangents  to  the  spheric  lines 
be  drawn,  AT  to  AEBF  and  in  its  plane,  and  AS  to 
AGBH  and  in  its  plane,  the  angle  TAS  is  equal  to  EOG, 
and  is  the  angle  between  the  spheric  lines. 

Therefore,  the  angle  between  two  tangents  drawn  to 
two  spheric  lines  at  their  point  of  intersection  is  the 
angle  between  the  spheric  lines. 

Cor.  2.  When  the  angle  EOG  is  a  right  angle,  OG  is 
normal  to  the  plane  of  AEBF,  and  Cr  is  a  pole  to  the 
circle  AEBF. 

Therefore,  two  spheric  lines  ar6  perpendicular  to  one 
another  when  one  of  them  passes  through  a  pole  of  the 
other ;  and  in  this  case  each  passes  through  both  poles  of 
the  other. 

217.  The  spheric  figure  AEBGA,  formed  by  two 
spheric  lines  between  their  points  of  intersection,  is  a 


THREE   PLANES.  203 

lune.  The  points  A  and  B  are  vertices  of  the  lune,  and 
the  angle  between  the  spheric  lines  forming  its  sides  is 
the  angle  of  the  lune. 

Evidently  every  lune  is  accompanied  by  an  opposite 
congruent  lune,  as  AEBGA  and  AFBHA;  and  any  two 
spheric  lines  divide  the  whole  spheric  surface  into  four 
lunes  which  are  congruent  in  opposite  pairs. 

We  have  the  analogous  case  in  plane  geometry,  where 
any  two  intersecting  lines  divide  the  whole  plane  into 
four  sections,  which,  although  extending  to  infinity,  may 
properly  be  said  to  be  congruent  in  opposite  pairs. 

It  will  be  seen  from  this  and  other  cases  that  the 
analogy  between  plane  and  spheric  geometry  is  descrip- 
tive rather  than  metrical  in  kind. 

Three  Planes  —  Section  of  Three-faced 
Corner  or  Trihedral  Angle  —  Spheric  Tri- 
angle. 

218.  Just  as  a  plane  section  of  any  corner  is  a  plane 
polygon  with  a  side  corresponding  to  and  given  by  each 
face  of  the  corner,  so  the  section  of  a  corner  by  a  sphere 
with  its  centre  at  the  vertex  of  the  corner  is  a  spheric 
polygon,  whose  sides  are  parts  of  spheric  lines,  and  the 
number  of  whose  sides  is  the  same  as  that  of  the  faces 
forming  the  corner. 

In  spheric  as  in  plane  geometry  the  most  important 
polygon  is  the  triangle. 

In  plane  geometry  three  given  lines  can  form  but  one 
triangle,  since  they  determine  at  most  but  three  points. 
But,  as  every  spheric  line  meets  every  other  spheric  line 


204 


SOLID   OR   SPATIAL  GEOMETRY. 


in  two  points,  three  spheric  lines,  which  are  not  concur- 
rent, determine  six  points,  and  these  may  combine  in 
threes  to  form  eight  spheric  triangles. 

Therefore,  any  three  non-concurrent  spheric  lines  divide 
the  surface  of  the  sphere  into  eight  triangles. 

219.  Let  AHA'G,  BIB' J,  and  CFCE  be  three  non- 
concurrent  spheric  lines.  They  meet  in  the  six  points 
A,  B,  C,  A',  B',  C,  of  which  A  is  opposite  A',  B  opposite 
B',  and  C  opposite  C. 

The  eight  determined  triangles  are 

ABC,  ABC,  AB'C,  A'BC, 

A'B'C,  A'B'C,  A'BC,  AB'C. 

Since  A  is  opposite  A',  etc.,  the  arc  AB  =  the  arc  A'B'. 
Similarly,  arc  BC  =  arc  B'C  and  arc  CA  =  arc  C'A'. 
Also,  as  ACC'A'  and 
ABA'B'  determine  two 
planes,  the  angle  at  A  is 
equal  to  the  angle  at  A'; 
and  so  also  the  angle  at  ^j 
B  is  equal  to  the  angle  at 
B',  and  the  angle  at  C  to 
the  angle  at  C.  And  thus 
the  opposite  triangles  ABC 
and  A'B'C  have  all  their 
parts  in  the  one  respec- 
tively equal  to  the  corre- 
sponding parts  in  the  other 
superposable.  For  taking  the  centre  as  a  point  of  refer- 
ence, ABC  and  A'B'C  are  in  opposite  orders  of  rotation. 


But  the  triangles  are  not 


SPHERIC   TRIANGLE.  205 

In  the  case  of  plane  triangles  we  could  invert  one  of 
them,  or  turn  it  over  in  the  plane,  and  then  superimpose 
them;  but  this  operation  is  clearly  impossible  in  a 
spheric  figure.  Spheric  triangles  related  in  this  manner 
are  symmetrical,  or  conjugate,  to  one  another,  and  they 
are  evidently  produced  by  the  sections  of  two  symmetri- 
cal three-faced  corners  having  a  common  vertex  at  the 
centre  (Art.  39). 

The  eight  triangles  are  symmetrical  or  conjugate  in 
opposite  pairs  as  follows : 

ABC  and  A'B^O,  ABO  and  A'B'C,  AB'C  and  A'BC, 
A'BC  and  AB'C. 

220.  Let  ABC  (Fig.  of  219)  be  a  spheric  triangle 
formed  by  section  of  the  three-faced  corner  0  •  ABC. 

The  angles  at  A,  B,  and  C,  whose  measures  are  respec- 
tively those  of  the  three  dihedral  angles  of  the  corner, 
are  called  the  angles  of  the  triangle,  and  are  usually 
denoted  by  A,  B,  C;  and  the  arcs  BC,  CA,  and  AB  are 
called  the  sides  of  the  triangle,  and  are  denoted  by  a,  b,  c. 

Here  two  views  confront  us. 

If  linear  units  are  to  be  introduced,  and  arcs  are  to  be 
considered  with  respect  to  length,  the  length  of  the  radius 
of  the  sphere  is  involved,  and  our  investigations  are  con- 
fined to  some  one  sphere  whose  radius  is  known  or  deter- 
minable with  reference  to  the  unit  of  measure  employed. 

On  the  other  hand,  if  our  operations  and  results  are 
to  have  no  reference  to  the  length  of  the  radius,  we  must 
take  for  the  side  of  the  spheric  triangle,  not  the  arc  itself, 
but  its  ratio  to  the  radius. 

This  ratio  is  an  angle,  a  face  angle  of  the  corner  which 
in  section  gives  the  spheric  triangle. 


206  SOLID   OR   SPATIAL   GEOMETRY. 

Spheric  geometry  then  ceases  to  have  any  distinct 
relation  to  the  sphere,  except  in  name,  involves  no  rela- 
tion of  length,  and  becomes  a  geometry  of  direction 
only. 

In  what  follows  we  shall  not  confine  ourselves  exclu- 
sively to  either  view,  but  shall  adopt  that  which  serves 
our  purpose  at  the  time. 

In  the  majority  of  applications,  however,  the  second 
one  is  the  only  view  that  can  be  adopted. 

Thus  in  applying  the  results  of  spheric  geometry  to 
the  visible  surface  of  the  heavens,  anything  like  a  linear 
unit  is  out  of  the  question. 

According  to  the  second  view,  a  spheric  triangle  con- 
sists of  six  parts,  all  angles. 

Three  alternate  parts,  called  the  angles  of  the  triangle, 
are  respectively  equal  in  measure  to  the  dihedral  angles 
of  a  three-faced  corner,  and  the  remaining  three,  called 
the  sides  of  the  triangle,  are  respectively  equal  to  the 
face  angles  of  the  same  corner. 

And  thus  all  the  relations  which  exist  between  the 
dihedral  angles  and  face  angles  of  a  three-faced  corner, 
exist  also  between  the  angles  and  sides  of  a  spheric 
triangle. 

The  adaptation  to  a  triangle  described  on  a  given 
sphere  is  easily  effected;  for  it  is  only  necessary  to 
express  the  sides  in  radians  and  then  multiply  the  result 
by  the  radius  of  the  sphere. 

221.  In  any  three-faced  comer  the  sum  of  two  face 
angles  is  greater  than  the  third  (Art.  35). 

Therefore,  the  sum  of  two  sides  of  a  spheric  trian- 
gle is  greater  than  the   third  side ;   and  the  difference 


SPHERIC   TRIANGLE.  207 

between  any  two  sides  of  a  spheric  triangle  is  less  than 
the  third  side  of  the  triangle. 

222.   Theorem.     The  shortest  path  from  one  point  to 
another,    on    the    surface    of    a 
sphere,  is  along  the  spheric  line 
through  the  points. 

Let  A  and  B  be  the  points,  and 
C  be  any  point  on  the  spheric  ^  )^~~^^ 
join  AB.  With  A  and  B  as 
poles  describe  circles  PCD  and 
QCE  to  pass  through  C.  Take  P,  any  point  on  the  circle 
PCD,  and  draw  the  spheric  arcs  AP,  and  PB  cutting  the 
circle  QGE  in  Q. 

Then  APB  is  a  spheric  triangle,  and 

AP+PB>AB.  (221) 

Therefore,  P  lies  without  the  circle  QCE,  and  the 
circles  PCD  and  QCE  touch  at  G. 

Now  let  ADEB  be  any  path  on  the  sphere  from  A  to 
B.  Then  the  path  from  ^  to  D  may  be  brought  to 
extend  from  ^  to  C  by  turning  the  circle  PCD  about 
its  centre  A,  until  D  comes  to  C.  In  a  similar  way,  the 
path  from  Bto  E  may  be  brought  to  extend  from  B  to  C. 

But  by  this  change  the  whole  path  is  shortened  by 
the  distance  DE.  That  is,  the  path  is  shortened  by 
making  it  pass  through  C,  a  point  on  the-  spheric  line 
through  A  and  B. 

In  like  manner,  each  part  of  the  path  is  shortened  by 
making  it  pass  through  some  arbitrary  point  on  the 
spheric  line  AB. 


208  SOLID   OR   SPATIAL   GEOMETRY. 

Hence  the  path  is  shortest  when  every  point  lies  on 
the  spheric  line  from  A  to  B. 

Cor.  1.  When  two  spheric  circles  touch,  the  spheric 
line  through  their  poles  passes  through  their  point  of 
contact. 

Cor.  2.  When  two  points  on  a  sphere  are  in  line  with 
the  centre,  an  indefinite  number  of  equal  shortest  paths 
may  be  drawn  from  one  point  to  the  other. 

223.  In  any  three-faced  corner,  considering  two  face 
angles  and  the  opposite  dihedral  angles,  the  greater  face 
angle  is  opposite  the  greater  dihedral  angle ;  and  con- 
versely, the  greater  dihedral  angle  is  opposite  the  greater 
face  angle  (Art.  40). 

Hence  in  any  spheric  triangle,  considering  two  sides 
and  the  opposite  angles,  the  greater  side  is  opposite  the 
greater  angle;  and  conversely,  the  greater  angle  is  oppo- 
site the  greater  side. 

Cor.  1.  If  a  spheric  triangle  has  two  equal  sides,  it 
has  two  equal  angles  ;  and  conversely,  if  it  has  two  equal 
angles,  it  has  two  equal  sides. 

Cor.  2.  The  order  of  magnitude  of  A,  B,  C,  the  angles 
of  a  spheric  triangle,  is  the  same  as  that  of  a,  6,  c,  the 
sides  of  the  triangle. 

Hence  spheric  triangles,  like  plane  ones,  are  equi- 
lateral, and  isosceles,  and  scalene. 

224.  When  the  three  face  angles  of  a  three-faced  cor- 
ner are  given,  the  dihedral  angles  also  are  given  (Art. 
41.  Cor.  2);    and  conversely,  when  the  dihedral  angles 


SPHERIC   TRIANGLE.  209 

are  given,  the  face  angle  also  are  given  (Art.  44.  Cor.  2). 
Therefore, 

1.  When  the  sides  of  a  spheric  triangle  are  given,  the 
angles  also  are  given,  and  all  the  parts  are  known. 

This  case  holds  also  for  plane  triangles. 

2.  When  the  three  angles  of  a  spheric  triangle  are 
given,  the  sides  also  are  given,  and  all  the  parts  are 
known. 

This  does  not  hold  for  plane  triangles,  and  this  funda- 
mental difference  between  spheric  and  plane  geometry  is 
due  to  the  fact  that  spheric  geometry  has  no  theory  of 
similar  figures,  a  theory  which  plays  so  important  a  part 
in  plane  geometry.  Similarity  requires  an  equality  of 
tensors,  and  therefore  involves  the  consideration  of  linear 
extension.  But  in  a  spheric  triangle,  where  all  the  parts 
are  angles,  there  is  no  place  for  linear  extension,  and 
hence  no  similarity  exists  beyond  absolute  equality. 

Similar  spheric  triangles  might  be  drawn  upon  spheres 
of  different  radii,  but  the  comparison  of  these,  although 
belonging  to  spatial  geometry,  does  not  belong  to  spheric 
geometry  (Art.  211). 

225.  In  any  corner  the  sum  of  the  face  angles  is  less 
than  a  circumangle  (Art.  42). 

Hence  the  sum  of  the  sides  of  a  spheric  triangle  is 
less  than  a  circumangle;  or  if  we  introduce  the  radius, 
is  less  than  a  circumference.     . 

Cor.  1.  When  two  sides  of  a  spheric  triangle  become 
straight  angles,  the  third  side  vanishes  and  the  figure 
becomes  a  lune. 

Cor.  2.     When  each  side  becomes  a  right  angle,  the 


210  SOLID   OR   SPATIAL   GEOMETRY. 

planes  forming  the  faces  of  the  corner  become  the  rec- 
tangular co-ordinate  planes  of  space  (Art.  8.  Cor.),  and 
each  angle  becomes  a  right  angle. 

Hence  a  spheric  triangle  which  has  each  side  a  right 
angle  has  also  each  angle  a  right  angle.  Such  a  triangle 
is  a  quadrantal  triangle. 

Cor.  3.  If  two  sides  of  a  spheric  triangle  be  produced 
to  meet,  a  second  triangle  is  formed  which  is  said  to  be 
co-lunar  with  the  first.  These  two  triangles  have  an 
angle  and  the  opposite  side  respectively  equal,  while  the 
remaining  two  sides  in  the  one  are  supplementary  to  the 
corresponding  sides  in  the  other,  and  the  remaining 
angles  in  the  one  are  supplementary  to  the  remaining 
angles  in  the  other. 

Cor.  4.  Any  spheric  triangle  has  three  colunar 
triangles. 

226.  The  polar  triangle.  When  the  vertices  of  one 
spheric  triangle  are  poles  to  the  sides  of  another  spheric 
triangle,  the  first  triangle  is  said  i  , 

to  be  polar  to  the  second.     And  ^^  \ 

the  second  triangle  is  also  polar  /V      \.  \ 

to  the  first.  o,/  \,^        1 

Let  A\  B',  C  be  poles  of  a,      /   /  \,,^^  \ 

b,  c  respectively,  and  let  a',  b',  c'  b( I 'Z^^ 

be  the  sides  of  the  spheric  tri-     \   1  ^^^^^-"^'^^    / 

angle  having  A',  B',  C  as  ver-  h--^'^  / 

tices.  / 

Since  C  is  pole  to  AB,  CO  is  1.  to  OB  (Art.  215) ; 
and  since  A'  is  pole  to  BC,  ^'0  is  ±  to  OB. 

Therefore,  OB  is  normal  to  the  plane  of  OA'  and  DC, 


SPHERIC   TRIANGLE.  211 

and  B  is  therefore  pole  to  the  spheric  join  of  A'C ;  that 
is,  to  b'. 

Similarly,  A  is  pole  to  a'  and  C  to  c';  and  the  spheric 
triangle  ABC  is  polar  to  A'B'C. 

Hence  when  one  spheric  triangle  has  its  vertices  poles 
to  the  sides  of  a  second  triangle,  the  vertices  of  the 
second  triangle  are  poles  to  the  sides  of  the  first,  and  the 
two  triangles  are  polar  to  each  other. 

Cor.  1.     A  quadrantal  triangle  is  polar  to  itself. 

Cor.  2.  The  points  A',  B',  C  determine  eight  triangles 
(Art.  218),  every  one  of  which  might  be  said  to  be  polar 
to  ABC.  Or  more  generally.  A,  B,  C  determine  one 
set  of  eight  triangles,  and  A',  B',  C  determine  a  second 
set;  and  every  triangle  of  one  set  has  every  triangle  of 
the  other  set  as  a  polar  triangle. 

It  is  easily  seen,  however,  that  two  triangles  in  either 
set  are  conjugate  (Art.  219),  and  the  remaining  six  are 
co-lunars  of  these.  So  that  if  we  agree  that  the  spheric 
triangle  formed  from  three  given  spheric  lines  is  to  be 
considered  as  being  the  triangle  (triangle  or  its  conjugate), 
each  of  whose  sides  are  less  than  a  straight  angle  or  a 
semi-circumference,  then  each  spheric  triangle  has  but 
one  polar  triangle  (a  triangle  or  its  conjugate). 

227.  The  spheric  triangles  ABC  and  A'B'C  being 
polar  to  one  another,  A'O  is  normal  to  the  plane  of  a, 
and  B'O  is  normal  to  the  plane  of  b.  But  the  angle 
between  two  planes  is  the  supplement  of  the  angle 
between  normals  to  the  planes  (Art.  16.  Def.  2);  and 
the  angle  between  the  planes  is  the  angle  C,  and  the 
angle  between  the  normals  is  the  side  c'. 
.-.  C+c' =  7r; 


212  SOLID   OR   SPATIAL   GEOMETRY. 

Or  the  sum  of  an  angle  of  a  spheric  triangle  and  the 
corresponding  opposite  side  of  the  polar  triangle  is  a 
straight  angle. 

Cor.    ^  +  a'=J5+6'=C+c'=^'  +  a=5'+&=C"+c=7r. 

228.  From  the  preceding  article, 

^  +  5  +  C  +  a'  +  6'  +  c'  =  Stt. 
But  (Art.  225)  a'  +  6'  +  c'  >  0  and  <  2  tt  ; 

.-.  A-\-B+G\^  < Stt  and  > tt. 

That  is,  the  sum  of  the  angles  of  a  spheric  triangle  is 
variable,  lying  between  the  limits  of  two  right  angles 
and  six  right  angles.  And  hence,  in  every  spheric  tri- 
angle the  sum  of  the  angles  exceeds  two  right  angles. 

The  amount  by  which  the  sum  of  the  three  angles 
exceeds  a  straight  angle  is  called  the  spherical  excess  of 
the  triangle.     If  we  denote  it  by  E,  we  have 

E  =  A  +  B+C-Tr. 

229.  It  has  been  shown  (Art.  219)  that  conjugate 
spheric  triangles,  although  having  all  their  corresponding 
parts  respectively  equal,  are  not  superposable,  but  corre- 
spond to  one  another  after  the  manner  of  the  right  and 
the  left  hand. 

Hence  in  the  determination  of  a  spheric  triangle  from 
its  parts,  there  is  always  the  kind  of  ambiguity  which 
results  from  not  knowing  whether  a  particular  triangle 
or  its  conjugate  is  the  one  required. 

This  ambiguity  disappears  in  the  case  of  an  isosceles 
spheric  triangle,  for  this  triangle  is  conjugate  to  itself. 


SPHERIC   TRIANGLE. 


218 


For  the  sake  of  simplicity,  then,  we  shall  agree  in 
what  follows  that  a  spheric  triangle  is  given  or  deter- 
mined when  we  know  the  triangle  or  its  conjugate. 

230.  A  spheric  triangle  is  given  when  the  three  sides 
are  given,  or  when  the  three  angles  are  given. 

This  follows  from  Art.  41.  Cor.  2,  and  Art.  224. 

231.  A  spheric  triangle  is  given  when  two  sides  and 
the  included  angle  are  given. 

Let  ABC,  A'B'C  be  two  spheric  triangles  in  which 
Z  C  =  Z  C,  a'  =  a,  and  b'  =  b;  and  let  the  parts  be  dis- 
posed in  the  same  order. 

Place  Con  0  and  C'A' 
along  CA.  Since  b'  =  b, 
A'  coincides  with  A.  Also, 
since  Z  C  =  Z  C,  the  side 
a'  will  lie  along  the  side 
a,  and  as  a'  =  a,  B'  will 
coincide  with  B. 

And  as  A  and  B  determine  only  one  spheric  line,  A'Bi 
coincides  with  AB,  and  the  triangles  coincide  in  all  their 
parts. 

Therefore,  the  triangle  ABC  is  given  when  two  sides 
and  the  included  angle  are  given  (see  P.  Art.  66). 

232.  A  spheric  triangle  is  given  when  two  angles  and 
the  included  side  are  given. 

Let  A  and  B  and  the  side  c  be  given. 

Then  the  sides  a'  and  b'  and  the  included  angle  C  is 
given  for  the  polar  triangle,  and  therefore  the  polar 
triangle  is  given  (Art.  231). 

Hence  the  original  triangle  is  given. 


214 


SOLID   OR   SPATIAL   GEOMETRY. 


233.   Let  ABC  be  an  isosceles  spheric  triangle  with 
CA  =  CB,  and  hence  the  ZA=  ZB. 

Draw  the  spheric  line  D'CD  to  the  middle  point,  D, 
of  the  base  AB.     Then  CD 
is  median  to  the  base. 

The  triangles  ACD  and 
BCD  having  AC  and  AD 
respectively  equal  to  BC  and 
BD,  and  the  included  angles 
A  and  B  equal,  are  conju- 
gate. 


Therefore, 


and 


ZCDA  =  ZCDB  =  1, 
Z  ACD  =  Z  BCD. 


Hence  the  median  to  the  base  of  an  isosceles  spheric 
triangle  is  the  right  bisector  of  the  base,  and  the  bisector 
of  the  vertical  angle. 

Hence,  also,  every  point  on  the  spheric  line  D'CD  is 
equidistant  from  A  and  B,  distance  being  measured  along 
a  spheric  line. 

In  the  same  manner  as  in  plane  geometry  (P.  Art.  54) 
it  is  shown  that  every  point  equidistant  from  A  and  B, 
and  lying  on  the  spheric  surface,  is  on  the  right 
bisector  of  the  base  AB,  i.e.  on  the  spheric  line 
D'CD. 

Cor.  In  spheric  geometry  the  join  of  A  and  B  is 
either  ADB  or  AD'B;  i.e.  there  are  two  joins  whose  sum 
makes  up  the  whole  spheric  line. 

The  bisector  of  one  of  these  joins  evidently  bisects 
the  other  also ;  as  at  Z>  and  D'. 


SPHERIC   TRIANGLE.  215 

234.  Let  P  (Fig.  of  233)  be  the  pole  of  AB,  and  let 
C  be  any  point,  lying  between  P  and  the  arc  AB,  on  the 
spheric  line  PD. 

Then    PA  =  PD,  and  PA  is  <  PC  +  CA,    (Art.  221.) 

and  .-.  CD  is  <  CA. 

And        Z  PAD  =  Z  PDA  =  1,  and  Z  CAD  is  <  1. 

Therefore,  the  least  distance  from  C  to  the  spheric  line 
AB  is  along  the  perpendicular  CD. 

Also,  two  equal  spheric  lines,  CA  and  CB,  can  be 
drawn  from  Cto  AB,  and  these  lie  upon  opposite  sides 
of  CD,  and  are  equally  inclined  to  it,  and  meet  the  line 
AB  at  equal  distances  from  the  foot  of  the  perpendicular. 

235.  From  C  draw  any  spheric  line,  CE,  to  meet  AB, 
and  to  cut  PA  at  some  point  F  between  P  and  A. 

Then,  PE  is  <PF+  FE,  (Art.  221.) 

and  GA\s<CF+FA; 

Therefore,  adding,  PE-{-CA<CE  +  PA. 

But  PE  =  PA; 

.:  CAis<CE. 

This  holds  true  so  long  as  CE  intersects  PA  between 
P  and  A.  But  as  E  recedes  from  A  along  the  spheric 
line  AB,  CE  will  continue  to  meet  PA  between  P  and 
A  until  E  comes  to  D\  opposite  D. 

Then  if  E  be  supposed  to  start  from  D  and  make  a 
complete  circuit  along  the  spheric  line  AB,  CE,  starting 
from  the  value  CD,  will  increase  as  DE  increases,  until 


216  SOLID   OR   SPATIAL   GEOMETRY. 

E  comes  to  D',  when  it  has  its  maximum  value.  It  then 
decreases  until  E  returns  through  B  to  D,  at  which 
position  CE  has  its  minimum  value. 

Cor.  1.  Since  all  spheric  lines  are  of  the  same  length, 
or  contain  the  same  angle,  the  greatest  and  least  spheric 
arcs  from  a  point  on  the  sphere  to  any  spheric  line,  AB, 
are  the  parts  of  the  spheric  line  perpendicular  to  AB, 
which  are  intercepted  between  G  and  AB. 

Cor.  2.  If  two  equal  spheric  arcs  be  drawn  from  a 
point  on  the  sphere  to  a  spheric  line  which  is  not  its 
equator,  they  are  equally  inclined  to  the  longest  spheric 
arc  from  the  given  point  to  the  given  line,  and  lie  upon 
opposite  sides  of  it.     . 

236.  As  in  a  plane  triangle,  so  in  a  spheric  one,  when 
two  sides  and  an  angle  opposite  one  of  them  are  given, 
the  triangle  may  be  ambiguous.  Owing  to  the  facts, 
however,  that  any  two  spheric  lines  intersect  in  two 
points,  and  that  the  sum  of  the  angles  of  a  spheric  tri- 
angle is  not  a  fixed  quantity,  the  condition  for  ambiguity 
is  much  more  complex  than  in  plane  geometry. 

Also,  unlike  a  plane  triangle,  a  spheric  triangle  may 
be  ambiguous  when  two  angles  and  a  side  opposite  one 
of  them  are  given. 

The  Ambiguous  Case. 

237.  In  the  following  examination  we  assume  that 
the  relative  magnitudes  of  the  parts  given  are  such  as 
to  determine  a  real  triangle,  so  that  we  shall  not  be 
concerned  with  conditions  which  lead  to  impossible  or 


THE  AMBIGUOUS   CASE.  217 

vanishing  triangles,  although  such  conditions  are  easily 
obtained. 

For  the  given  parts  let  us  take  the  sides  a  and  b  and 
the  angle  A,  and  let  us  consider  the  subject  under  three 
cases,  according  as  A  is  less,  equal  to,  or  greater  than, 
a  right  angle. 

For  a  general  diagram  let  ACD  and  AED  be  two 
spheric  lines  at  right  angles,  and  let  C  be  a  pole  of 
AED.  Through  A  and  D  draw  the  spheric  line  APD, 
making  the  angle  CAP  less  than  a  right  angle,  and  the 
spheric  line  AQD,  making  the  angle  CAQ  greater  than 
a  right  angle. 

Take  any  point  C"  between  ^ -^^^ 

A  and  C,  and  any  point  C"         cj/^....      p  >^ 

between  C  and  D.  h/y^'  ^^^^P\\ 

Let  the  side  b  be  measured     //  >\ 

from  A  along  the  arc  ACD,   J  _-^^° 

and  let  the  given  angle  A     W~~       y 1        "^^"^  J\ 

be  the  angle  at  A.     Then  a       \""->^^  ^^^Q  / 

is   drawn   from   some   point        \    Q"  q  / 

on  the  arc  ACD  to  the  arc  X.,.^^         ^^^ 

APD    or    AED    or    AQD, 

according  as  A  is  less  than  a  right  angle,  equal  to  a  right 
angle,  or  greater  than  a  right  angle.  One  such  triangle 
is  represented  at  AC'P,  where  a  denotes  the  side  C'P. 

Case  I.  Let  ^  =  Z  CAP,  <  f- 

Then  the  shortest  distance  from  a  point  on  ACD  to 
the  arc  APD  is  perpendicular  to  APD  (Art.  234),  and 
is  therefore  not  along  ACD. 

1.  Let  b  =  AC,  <  f ,  and  let  P'  be  the  foot  of  the 
perpendicular  from  C. 


218  SOLID   OR   SPATIAL   GEOMETRY. 

Two  equal  arcs  can  be  drawn  from  C  to  APD,  one  on 
each  side  of  C'P'  (Art.  234),  and  the  triangle  may  be 
ambiguous. 

For  a  case  of  ambiguity,  however,  a  must  be  <  C'A, 
i.e.  <  6. 

.-.  -4  <  f,  6  <  f,  a  <  6  is  ambiguous. 

2.  Let  h  =  AC=%  and  let  P  be  the  foot  of  the  per- 
pendicular from  C. 

Then  evidently  the  case  is  ambiguous  if  a  is  less  than 
AC  and  greater  than  CP,  and  CP,  being  on  the  equator 
to  A,  measures  the  angle  A  (Art.  216). 

■>  A 

.'.  -4<  f,  6  =  f,  a^  ,   is  ambiguous. 

3.  Let  h  =  AC",  >  f .  Then  the  case  will  be  ambiguous 
if  a  <  C"D. 

.-.  -4<5,  6>f,  a<(7r  — 6)  is  ambiguous. 

Cor.  In  all  the  foregoing  cases  it  is  readily  seen  that 
the  ambiguity  disappears  when  the  triangle  becomes 
right  angled  by  a  being  drawn  perpendicular  to  the  arc 
APD. 

Case  II.   Let  A  be  the  angle  CAE  =  f . 

Since  C  is  a  pole  of  AED,  from  any  point  on  ACD, 
two  equal  spheric  arcs  can  be  drawn  to  AED,  one  lying 
on  each  side  of  AC  and  equally  inclined  to  it  (Art. 
234).  The  two  triangles  thus  formed  would  be  conju- 
gate and  not  ambiguous  (Art.  229).  But  if  the  point 
C  be  taken,  all  spheric  arcs  from  C  to  AED  are  equal, 
and  the  triangle  is  indeterminate. 

Therefore,  with  A  =  ^  there  is  no  real  ambiguity, 
but   when   6  <  f   and  a  >  6,  and  also  when  6  >  f   and 


THE  AMBIGUOUS   CASE.  219 

a>(7r  — 6),  two  triangles  are  obtained  which  are  conju- 
gate ;  and  when  6  =  f ,  the  triangle  is  indeterminate. 

Case  III.  Let  A  be  the  angle  CAQ>^.  Since  the 
angle  CAQ  is  greater  than  a  right  angle,  the  longest 
spheric  arc  from  any  point  on  ACD  to  AQD  is  perpen- 
dicular to  AQD,  and  is  therefore  not  along  ACD. 

1.  Let  b  =  AC',  <|;  and  let  Q'  be  the  foot  of  the 
perpendicular  from  C  to  AQD. 

Then  two  equal  spheric  segments  may  be  drawn  from 
C  to  the  arc  AQD,  one  on  each  side  of  C'Q'  (Art.  235. 
Cor.  2),  and  the  triangle  may  be  ambiguous.  For  a  case 
of  ambiguity,  however,  C'Q'  must  be  greater  than  CD, 
or  a>  (tt  —  6). 

.'.  A>1,  6<f,  a>(7r  — 6)  is  ambiguous. 

2.  Let  b  =  AC=l;  then  if  Q  be  the  foot  of  the  per- 
pendicular from  C,  it  is  readily  seen  that  the  case  will 
be  ambiguous  if  a  is  less  than  CQ  and  greater  than  CD. 
But  CQ  being  on  the  equator  to  A  measures  the  angle 
A  (216). 

<  A 

.'.  -4  >  f ,  6=1,  « v^  J,  is  ambiguous. 

3.  6  =  AC",  >  f .  Then,  if  Q"  be  the  foot  of  the 
perpendicular  from  C",  the  triangle  will  be  ambiguous  if 
a  lies  between  C'Q"  and  C"A. 

.-.  A>1,  6  >  ?,  a  >  6  is  ambiguous. 


220 


SOLID   OR   SPATIAL   GEOMETRY. 


238.   The  results  of  the  preceding  article  are  collected 
in  the  following  table : 


6<f 

6  =  1 

6<  f 

A<f 

a<6 
ambiguity. 

a  >^<6 

ambiguity. 

a  <  (tt  -  6) 
ambiguity. 

A  =  l 

a  >  6  <  (ir  -  6) 
conjugate. 

a  —  h 
indeterminate. 

a  <  6  >  (ir  -  6) 
conjugate. 

A>1 

a  Xtt  -  6) 
ambiguity. 

ffl  <  ^>  6 

ambiguity. 

ambiguity. 

We  see  from  the  table  that  ambiguity  occurs  only 
when  A  is  not  a  right  angle. 

239.  By  making  use  of  the  polar  triangle  we  can 
readily  investigate  the  cases  of  ambiguity  when  two 
angles  and  a  side  opposite  one  of  them  are  given.  For 
when  a  triangle  is  ambiguous,  its  polar  is  ambiguous,  and 
vice,  versa. 

The  table  corresponding  to  that  of  the  last  article  is 
here  given : 


5>f 

B  =  ^ 

-B<f 

«>  2 

A>B 

ambiguity. 

A<a>B 

ambiguity. 

A>iw-B) 
ambiguity. 

a  =  l 

A<B>(_',r-B) 
conjugate. 

A  =  B 

indeterminate. 

A>B<(x-B) 

conjugate. 

a<5 

A<(^-B) 
ambiguity. 

A>a<B 

ambiguity. 

A<B 

ambiguity. 

THEOREMS. 


221 


240.  Theorem.  In  any  right-angled  spheric  triangle 
the  number  of  sides  which  are  less  than,  equal  to,  or 
greater  than,  a  right  angle  is  at  least  two. 

In  the  figure  of  Art.  237,  let  ^  be  a  right  angle. 

1.  If  6  =  f ,  then  a  =  f ,  and  two  sides  are  right  angles. 

2.  Let  b  =  AC,  and  let  E  be  the  pole  of  ACD.  Then, 
C'E=  5,  and  C'f  is  <  f,  and  G'e  is  >  f.  But,  also,  Af  is 
<  f ,  and  Ae  is  >  f . 

And  a  and  c  are  both  less,  or  equal  to,  or  greater 
than,  f . 

3.  Let  6  =  AC".  Then  &  >  f ;  and  c"e  is  <  f ,  and 
C"f  is  >  f.     But  Ae  is  >  f,  and  Af  is  <  f. 

Therefore,  when  6  <  f ,  both  a  and  c  are  less  than, 
equal  to,  or  greater  than,  f . 

When  6  =  i,  both  a  and  b  are  equal  to  f . 

When  6  >  f ,  &  and  c  are  >  f ,  or  a  and  c  are  =  f ,  or 
6  and  a  are  >  f . 

And  the  theorem  is  proved. 

241.  Theorem.  The  sum  of  two  angles  of  a  spheric 
triangle,  and  the  sura  of  the  sides  opposite  these  angles, 
are  both  less  than,  equal  to,  or  greater 
than,  a  straight  angle. 

Let  CDF  be  a  lune,  and  let  EF  be 
equator  to  C  and  D. 

Through  G,  the  middle  point  of  EF, 
draw  the  spheric  line  AB,  meeting  the 
sides  of  the  lune  in  A  and  B.  Then 
CAB  is  a  spheric  triangle. 

It  is  evident  that  the  triangles  CAB 
and  DBA  are  congruent,  and  therefore 
that  Z  GAB  =  Z  DBA,  and  Z  CBA  =  Z  DAB,  etc. 


222  SOLID    OR    SPATtAL   GEOMETRY. 

Then  Z  CAB  +  Z  CBA  =  Z  CAB  +  Z  BAD  =  tt, 
and  CA  -\- CB  =  CA -i- AD  =  TT. 

Therefore,  if  the  sum  of  two  angles  is  a  straight  angle, 
so  also  is  the  sum  of  the  two  opposite  sides. 

Now  take  B',  any  point  between  B  and  Z>,  and  draw 
the  spheric  arc  AB'. 
Then 
Z  ABB'  +  Z  BAB'  +  ZBB'A  >  tt;    (Art.  43.  Cor.  2.) 
and  ZABB'  =  ZCAB. 

.-.  ZCAB' +  ZBB' A  >7r. 
Also,  CA  +  CJB'  >  CA  4-  0-B  >  TT. 

Therefore,  when  the  sum  of  two  angles  is  greater  than 
a  straight  angle,  the  sum  of  the  opposite  sides  is  also 
greater  than  a  straight  angle. 

And  since  these  sums  decrease  and  increase  together, 
the  theorem  follows. 

Hence  ^(A-\-B)  and  |(a4-&)  are  both  >  f ,  both 
=  f,  or  both  <  f. 

This  relation  is  commonly  expressed  by  saying  that 
^{A-\-  B)  and  ^{a  -\-b)  are  of  the  same  affection. 

EXERCISES   Q. 

1.  The  area  of  a  spheric  triangle  is  Er^,  where  ^  is  the  spherical 

excess. 

2.  The  area  of  a  spheric  polygon  is  {'EA  —  (n  —  2)ir}r'^,  where 
2.4  is  the  sum  of  the  angles,  and  n  is  the  number  of  sides. 

3.  The  area  of  an  equilateral  spheric  triangle  is  one-fourth  that 
of  the  surface  of  the  sphere.  Show  that  its  angle  is  120°,  and  find 
its  side. 


EXERCISES.  223 

4.  ^B  is  a  spheric  arc,  and  C  is  its  middle  point.  The  locus  of 
P,  such  that  Z  APC  =  Z  BPC,  is  two  spheric  lines  perpendicular 
to  one  another.  Prove  this,  and  state  its  analogue  in  plane 
geometry. 

6.  If  the  direction  from  A  to  B,  two  places  on  the  earth,  is 
estimated  along  a  spheric  line,  and  in  terms  of  the  angle  which  this 
line  makes  with  the  meridian  of  the  first  place,  show  that  if  A  and 
B  have  different  latitudes  and  longitudes  the  direction  from  Ato  B 
is  not  the  opposite  of  the  direction  from  Bto  A. 

6.  If  a  spheric  triangle  be  formed  by  cutting  a  three-faced 
corner  by  a  sphere,  the  centre  of  the  sphere  being  the  vertex  of  the 
comer,  show  (i.)  that  the  isoclinal  line  to  the  edges  of  the  corner 
gives  the  centre  of  the  circle  circumscribing  the  spherical  triangle  ; 
(it.)  That  the  isoclinal  line  to  the  faces  gives  the  centre  of  the 
inscribed  circle  of  the  triangle. 

7.  "What  are  given  by  the  external  isoclinal  lines  to  the  comer  ? 

8.  A  spheric  line  is  described  by  a  quadrant  which  has  one 
extremity  fixed  (215)  ;  what  is  the  analogue  in  plane  geometry  ? 


224  SOLID   OK   SPATIAL   GEOMETRY. 


MISCELLANEOUS  EXERCISES. 

1.  Non-parallel  lines  do  not  necessarily  intersect. 

2.  Two  circles  in  space  may  pass  within  one  another  and  have 
two,  one,  or  no  points  in  common. 

3.  From  the  definition  of  a  tangent  (P.  Art.  109)  show  that 
the  tangent  to  a  circle  lies  in  the  plane  of  the  circle. 

4.  A  plane  which  is  normal  to  the  common  line  of  two  planes 
is  perpendicular  to  both  planes. 

5.  If  any  number  of  planes  meet  in  parallel  lines,  the  normals 
to  these  planes,  from  the  same  point,  are  complanar. 

6.  The  sum  of  the  normals  from  a  point  A  to  the  planes  U  and 
V  is  the  same  as  that  of  the  normals  from  B  to  the  same  planes. 
Show  that  if  P  be  any  point  in  the  line  AB,  the  sum  of  the  normals 
from  Pto  U  and  V  is  constant. 

7.  Show  that  Ex,  6  holds  good  for  any  number  of  planes,  U, 
V,  W,  etc. 

8.  If  the  sum  of  the  normals  to  the  planes  U  and  V  be  the 
same  for  any  three  points,  A,  B  and  C,  it  is  the  same  for  every 
point  in  the  plane  of  ABC. 

9.  The  right-bisector  plane  of  the  common  perpendicular  to  two 
lines  bisects  the  join  of  any  two  points,  one  on  each  line. 

10.  A  perpendicular  is  drawn  to  the  base  of  a  regular  pyramid 
and  meets  the  faces,  produced  where  necessary.  Then,  the  sum  of 
the  distances  of  the  points  of  intersection  from  the  base  is  constant. 

11.  Find  in  a  given  plane,  a  point  equidistant  from  three  given 
points. 

12.  'Determine  on  a  given  line  the  point  which  is  equidistant 
from  any  two  given  points. 

13.  The  bisecting  plane  of  a  dihedral  angle  of  a  tetrahedron 
divides  the  opposite  edge  into  segments  which  are  proportional  to 
the  areas  of  adjacent  faces. 

14.  The  shortest  chord  through  any  point  within  a  sphere  is 
normal  to  the  diametral  plane  containing  the  point. 


MISCELLANEOUS  EXERCISES.  225 

15.  If  one  intersection  of  a  sphere  by  a  cylinder  is  a  circle,  so 
also  is  the  other  intersection. 

16.  If  one  intersection  of  a  sphere  by  a  cone  is  a  circle,  so  also  is 
the  other  intersection. 

17.  The  perpendiculars  to  the  faces  of  a  tetrahedron,  at  their 
centroids,  are  concurrent. 

18.  The  vertices  of  a  cuboid  are  conspheric. 

19.  The  edges  of  a  given  three-faced  corner  pass  through  three 
fixed  points.     Show  that  its  vertex  is  fixed. 

20.  The  medians  of  a  tetrahedron,  taken  in  both  length  and 
direction,  form  a  quadrilateral. 

21.  In  a  three-faced  corner  the  planes  through  the  edges  bisect- 
ing the  dihedral  angles  form  an  axial  pencil. 

22.  In  a  three-faced  corner  the  planes  through  the  bisectors  of 
the  face-angles  and  perpendicular  to  the  faces  form  an  axial  pencil. 
How  many  such  pencils  in  all  ? 

23.  What  is  the  locus  of  a  point  equidistant  from  two  given 
points  ? 

24.  What  is  the  locus  of  a  point  equidistant  from  two  complanar 
lines  ? 

25.  What  is  the  locus  of  a  point  equidistant  from  two  given 
planes  ? 

26.  What  is  the  locus  of  a  point  equidistant  from  three  parallel 
lines  ? 

27.  A  point  is  equidistant  from  a  fixed  line  and  a  fixed  plane  ; 
show  that  its  locus  is  a  ruled  surface. 

28.  Through  a  given  line  pass  a  plane  perpendicular  to  a  given 
plane. 

29.  Through  a  given  point  pass  a  plane  normal  to  a  given  line. 

29.}.  The  locus  of  a  point  whose  joins  with  two  given  points  is 
in  a  constant  ratio  is  a  sphere  whose  centre-line  passes  through  the 
given  points. 


226  SOLID   OR   SPATIAL  GEOMETRY. 

30.  Given  a  plane  and  any  three  points,  show  that  a  point  may 
be  found  in  the  plane  such  that  its  joins  with  the  given  points  shall 
make  equal  angles  with  the  plane. 

31.  Through  a  given  point  draw  a  line  to  intersect  two  given 
non-complanar  lines. 

32.  Through  a  given  point,  P,  in  a  plane  draw  a  planar  line 
which  shall  be  at  a  given  distance  from  a  given  point,  Q.  What 
are  the  limits  of  possible  solution  ? 

33.  Draw  a  line  from  a  point,  P,  to  a  plane,  M,  which  shall  be 
parallel  to  the  plane  N,  and  of  given  length. 

34.  Given  L,  M,  N,  three  non-complanar  lines,  draw  a  line  to 
intersect  L  and  M  and  be  perpendicular  to  JV.    To  be  parallel 

toiV. 

35.  Through  a  given  point  to  draw  a  line  which  shall  meet  a 
given  line  and  a  given  circle  not  complanar  with  the  line. 

36.  Two  points  are  upon  opposite  sides  of  a  plane.  Find  the 
point  in  the  plane  for  which  the  difference  of  its  distances  from  the 
given  points  shall  be  a  maximum. 

37.  In  Ex.  36  find  a  point  in  the  plane  which  shall  be  equidis- 
tant from  the  given  points. 

38.  Cut  a  given  four-faced  corner  by  a  plane  so  that  the  section 
shall  be  a  parallelogram. 

39.  L  and  M,  two  non-complanar  lines,  meet  their  common  per- 
pendicular in  A  and  B.     If  P  be  any  point  on  L,  and  Q  on  M, 

P§2  =  AB'-  +  AP-^  -f-  P§2  -2APBQ  cos  6, 

where  0  is  the  angle  between  the  lines  L  and  M. 

40.  O  is  the  centre,  e  an  edge,  and  A  a  vertex  of  a  ppd. ,  and  P 
is  any  point.     Then,  SP^l^  -  8  PO^  +  ^  Se^. 

41.  O  is  the  centroid,  and  a  is  a  side  of  any  triangle,  and  P  is 
any  point  in  space.    Then,  I,PA^  -  3  PO'-  +  \  'LalK 

42.  0  is  the  centre,  ^  is  a  vertex,  and  e  is  an  edge  of  a  tetrahe- 
dron, and  P  is  any  point.    Then,  'Z.PA^  =  4  PO^  +  \  Se^, 


MISCELLANEOUS   EXERCISES.  227 

43.  From  a  fixed  point  three  mutually  perpendicular  lines  are 
drawn  to  a  fixed  plane.  Show  that  the  sum  of  the  squares  of  the 
reciprocals  of  these  lines  is  constant,  being  the  square  of  the  recip- 
rocal of  the  perpendicular  from  the  point  to  the  plane.  (Compare 
P.  Ex.  32,  p.  131.) 

44.  In  any  four-sided  prism  the  sum  of  the  squares  on  the 
twelve  edges  is  greater  than  the  sum  of  the  squares  on  the  four 
diagonals,  by  eight  times  the  square  on  the  join  of  the  common 
mid-points  of  the  diagonals,  taken  in  pairs. 

45.  What  are  the  axes  of  symmetry  of  a  cube?  Of  a  cuboid? 
Of  a  regular  tetrahedron  ? 

46.  Two  spheres  may  have  two,  one,  or  no  common  tangent 
cones.  Distinguish  the  cases,  and  explain  those  where  the  spheres 
have  contact  of  the  same  and  of  opposite  kinds. 

47.  Of  four  spheres,  each  one  touches  three  others.  Show  that 
their  tangent  planes,  at  the  points  of  contact,  form  a  sheaf  of 
planes. 

48.  The  common  tangent  cones  to  three  spheres,  taken  in  twos, 
have  their  centres  in  four  collinear  rows  of  three. 

49.  The  common  tangent  cones  to  four  spheres,  taken  in  twos, 
have  their  centres  lying  by  sixes  upon  four  planes. 

60.  Two  circles  in  parallel  planes  are  the  intersections  of  the 
planes  by  two  different  cones.  Show  that  the  centres  of  the  cones 
and  the  centres  of  the  circles  form  a  harmonic  range. 

51.  The  difference  between  two  faces  of  a  tetrahedron  is  less 
than  the  sum  of  the  other  two. 

52.  Show  that  to  bisect  a  pyramid  by  a  plane  parallel  to  the  base 
requires  the  solution  of  a  cubic  equation. 

53.  The  cube  having  the  diagonal  of  another  cube  for  its  edge 
has  3  y/3  times  the  volume  of  the  other. 

54.  One  cube  has  its  face  equal  to  the  surface  of  another  cube. 
Compare  their  volumes,  and  also  their  edges. 


228  SOLID   OR   SPATIAL   GEOMETKY. 

66.  If  P  be  any  point  within  a  parallelepiped  whose  diagonals 
are  A4',  BB\  etc.,  the  pyramids 

PABCD  +  P-  A'B'C'D'  =  ^  the  ppd. 
What  if  P  be  without  ? 

66.  If  a,  b,  c,  d  be  the  four  altitudes  of  a  tetrahedron,  and 
a',  &',  c',  d'  be  the  corresponding  perpendiculars  from  any  point  to 
the  faces,  show  that 

^  +  ^  +  «'  +  ^  =  l. 
abed 

87.  ABGD  is  a  tetrahedron,  and  P  is  any  point.  If  AP,  BP, 
etc.,  meet  the  faces  in  a,  b,  etc.,  then 

Pa  .  Pb      Pc     Pd  _y^ 
Aa     Bb      Cc     Dd 

58.  Three  mutually  perpendicular  lines  pass  through  a  fixed 
point  in  a  sphere.  Show  that  the  sum  of  the  squares  of  the  three 
determined  chords  is  constant. 

59.  In  Ex.  58,  the  sum  of  the  squares  on  the  six  segments  into 
which  the  chords  are  divided  by  the  point,  is  constant. 

60.  A  spherical  shell  six  inches  in  diameter  has  the  interior 
cavity  one-half  the  volume  of  the  sphere.  Find  the  thickness  of 
the  shell. 

61.  Three  equal  spheres  touching  each  other  lie  upon  a  table, 
and  a  fourth  equal  sphere  rests  upon  the  three.  How  far  is  the 
centre  of  the  fourth  from  the  table  ? 

62.  In  Ex.  61,  the  radius  of  the  fourth  sphere  is  n  times  that  of 
the  others.     What  is  the  case  when  n  =  1  —  |  V3  ? 

63.  A  sphere  touches  each  of  three  mutually  perpendicular  con- 
current lines.  Find  the  distance  from  the  centre  of  the  sphere  to 
the  point  of  concurrence. 

A  cylinder  of  revolution  whose  section  througli  the 
axis  is  a  square  is  an  equilateral  cylinder;  and  the  cone 
of  revolution  whose  section  through  the  axis  is  an  equi- 
lateral triangle  is  an  equilateral  cone. 


MISCELLANEOUS   EXERCISES.  229 

64.  If  an  equilateral  cone  and  an  equilateral  cylinder  be  in- 
scribed in  the  same  sphere, 

(1)  The  surface  of  the  cylinder  is  a  mean  proportional  between 
the  surfaces  of  the  sphere  and  cone  ; 

(2)  The  volume  of  the  cylinder  is  a  mean  proportional  between 
the  volumes  of  the  sphere  and  cone. 

65.  If  an  equilateral  cone  and  an  equilateral  cylinder  be  circum- 
scribed about  the  sphere, 

(1)  The  surface  of  the  cylinder  is  a  mean  proportional  between 
the  surfaces  of  the  sphere  and  cone  ; 

(2)  The  volume  of  the  cylinder  is  a  mean  proportional  between 
the  volumes  of  the  sphere  and  cone. 

If  P,  Q  be  points  on  a  centre  line  of  a  sphere,  such 
that  OP  •  0Q  =  R-,  where  0  is  the  centre  of  the  sphere 
and  R  is  the  radius,  the  points  P  and  Q  are  inverse 
points  with  respect  to  the  sphere.  And  when  two 
figures  are  such  that  every  point  in  the  one  is  the  in- 
verse of  a  corresponding  point  in  the  other,  the  figures 
are  inverse  to  one  another  (P.  Art.  260) . 

66.  The  inverse  of  a  line  is  a  complanar  circle  through  the  centre 
of  inversion. 

67.  The  inverse  of  a  circle  is  a  complanar  circle,  unless  the  first 
circle  passes  through  the  centre  of  inversion. 

68.  The  inverse  of  a  sphere  is  a  sphere,  unless  the  first  sphere 
passes  through  the  centre  of  inversion,  when  its  inverse  is  a  plane. 

69.  The  inverse  of  a  plane  is  a  sphere  through  the  centre  of 
inversion. 

70.  A  sphere  which  passes  through  a  pair  of  inverse  points  with 
respect  to  another  sphere  cuts  the  other  orthogonally. 

71.  A  sphere  which  cuts  two  spheres  orthogonally  has  its  centre 
on  the  radical  plane  of  the  two. 


230  SOLID   OR   SPATIAL   GEOMETRY. 

72.  A  cube  circumscribed  to  a  sphere  is  inverted  with  respect  to 
the  sphere.  Show  that  the  spheres  produced  pass  by  threes  through 
common  points  and  cut  one  another  orthogonally. 

73.  The  locus  of  a  point  with  respect  to  which  two  spheres  can 
be  inverted  into  equal  spheres  is  a  sphere  having  a  common  radi- 
cal plane  with  the  two. 

74.  The  locus  of  a  point  with  respect  to  which  three  spheres  can 
be  inverted  into  equal  spheres  is  a  circle. 

75.  There  are  two  points,  real  or  imaginary,  with  respect  to 
which  four  spheres  can  be  inverted  into  equal  spheres. 

76.  What  is  the  locus  of  a  point  from  which  two  given  spheres 
subtend  the  same  angle  ? 

77.  What  is  the  locus  of  a  point  from  which  three  spheres  sub- 
tend the  same  angle  ? 

78.  The  joins  of  the  foci  to  any  point  on  a  hyperbola  are  equally 
inclined  to  the  tangent  at  that  point. 

79.  If  an  ellipse  and  a  hyperbola  have  the  same  foci,  the  curves 
intersect  orthogonally. 

80.  A  sector  of  a  circle  revolves  about  a  diameter  parallel  to  the 
chord  of  the  sector.  The  volume  described  is  ^irr^  sin  6,  where  2  6 
is  the  angle  of  the  sector. 

81.  The  volume  of  a  segment  of  a  sphere  is 

I  irr^  {2  -  3  cos  .^  +  cos2  ^}, 
where  2  ^  is  the  angle  subtended  by  the  segment. 

82.  A  plane  figure,  invariable  in  form  and  dimensions,  moves 
with  its  centre  on  a  path  which  is  inclined  to  its  plane  at  a  constant 
angle,  a.  Show  that  the  volume  described  is  the  area  of  the  figure 
X  the  length  of  path  x  sin  o. 

83.  The  generator  of  Art.  143  does  not  preserve  its  orientation, 
but  revolves  about  the  path.  Show  that  this  does  not  affect  the 
volume  described,  if  the  centroid  is  confined  to  the  path. 


MISCELLANEOUS   EXERCISES.  231 

84.  A  square,  side  s,  moves  with  its  centre  on  a  circle,  and  its 
plane  perpendicular  to  the  path,  but  revolves  about  the  path  as  an 
axis.     Show  that  the  volume  described  is  2  irrs-,  if  r  >  ^  sV2. 

85.  A  spheric  line  is  described  by  a  quadrant  rotating  about  one 
of  its  end-points  as  a  centre  (Art.  215).  What  is  the  analogue  in 
plane  geometiy  ? 

A  plane  through  one  of  two  inverse  points,  normal  to 
the  join  of  the  points,  is  the  polar  plane  to  the  other 
point,  and  this  latter  point  is  the  pole  of  the  plane,  with 
respect  to  the  sphere  of  inversion. 

86.  If  the  point  P  lies  on  the  polar  plane  of  Q,  then  Q  lies  on 
the  polar  plane  of  P. 

87.  The  polar  of  a  line  is  a  line  at  right  angles  to  the  given  line. 

88.  Explain  how  the  process  of  Art.  63  is  one  of  polar  recipro- 
cation. 

89.  Show  that  the  tetrahedron  may  be  a  polar  reciprocal  to 
itself. 

90.  A  sphere  touches  the  twelve  edges  of  a  cube.  What  is  the 
polar  reciprocal  of  the  cube  with  reference  to  the  sphere,  and  how 
is  it  situated  ? 

91.  The  distances  of  any  two  points  from  a  polar  centre  are  pro- 
portional to  the  distances  of  each  point  from  the  polar  plane  of  the 
other. 

92.  The  centre  locus  of  a  sphere  which  cuts  two  given  spheres 
orthogonally  is  their  radical  plane. 

93.  The  centre  locus  of  a  sphere  which  cuts  three  spheres 
orthogonally  is  their  radical  line. 

94.  All  the  spheres  which  cut  two  spheres  orthogonally  pass 
through  two  fixed  points. 

95.  All  the  spheres  which  cut  three  given  spheres  orthogonally 
pass  through  three  fixed  points. 


232  SOLID   OR   SPATIAL   GEOMETRY. 

96.  A  sphere  which  cuts  four  spheres  orthogonally  is  fixed. 
What  exception  ? 

97.  All  the  spheres,  which  have  contact  of  like  kind  (P,  Art. 
291)  with  two  given  spheres,  are  cut  orthogonally  by  one  and  the 
same  sphere. 

98.  All  the  spheres  which  have  contact  of  like  kind  with  three 
given  sjAeres  are  cut  orthogonally  by  the  same  three  spheres. 

99.  A  line  is  cut  harmonically  by  a  point,  a  conic,  and  the  polar 
of  the  point  with  respect  to  the  conic. 

100.  A  line  is  cut  harmonically  by  a  point,  a  sphere,  and  the 
polar  plane  of  the  point  with  respect  to  the  sphere. 

In  crystals,  wlietlier  formed  in  the  laboratory  or  by 
slow  geological  processes,  we  have  examples  of  natural 
polyhedra.  These  are  forms  derived  from  prisms  or 
parallelepipeds  by  transformations  closely  allied  to  polar 
reciprocation,  the  replacement  of  corners  or  points  by 
planes.  In  crystallography  the  relative  direction  of  the 
plane  which  forms  a  face  of  the  crystal  is  of  primary 
importance ;  its  distance  from  the  centre  is  only  a  sec- 
ondary consideration. 

Through  the  centre  of  the  cube  let  the  three  rectan- 
gular axes  of  space  be  drawn  parallel  to  the  direction 
edges  of  the  cube,  and  let  them  be  denoted  by  X,  Ysmd 
Z.  Every  plane  cuts  these  axes  either  at  finite  points 
or  at  infinity,  and  hence  every  plane  makes  on  these 
axes  three  intercepts,  which  may  be  finite  or  infinite. 

Denote  the  intercepts  by  x,  y,  z,  where  these  letters 
denote  measures  on  the  respective  axes,  but  may  be 
equal  or  unequal  in  value.  The  giving  of  these  inter- 
cepts determines  the  relative  direction  of  the  plane. 

If  a  plane  which  forms  a  face  of  a  crystal  is  parallel 
to  the  face  of  the  original  cube,  it  is  looked  upon  as  a 


MISCELLANEOUS    EXERCISES.  233 

face  of  the  cube,  and  if  parallel  to  a  face  of  the  regular 
derived  octahedron,  it  is  considered  to  be  a  face  of  the 
octahedron,  etc. 

101.  Show  that  the  plane  (x,  ?/,  z)  is  parallel  to  the  plane  (mx, 
wiy,  mz). 

102.  Show  that  the  plane    (x,  y,  z)   is  parallel  to  the  plane 

(?,   ^,   1).     AVhat  inference  do  you  draw  as  to  the  absolute  values 

\z    z       ) 

of  the  intercepts  ? 

103.  Show  that  the  plane  (x,  y,  —  z)  is  parallel  to  the  plane 
(-X,  -y,z'). 

104.  The  planes  (2,  1,  1)  and  (—  ^,  1,  1)  are  perpendicular  to 
one  another. 

105.  The  planes  (a,  a,  6)  and  f  «,  a,  —  —  )  are  perpendicular  to 
one  another.  ^  ' 

106.  Show  that  (1,  oo  ,  oo  )  is  a  cubic  face,  and  write  the  remain- 
ing faces  of  the  cube. 

In  representing  planes  in  this  way,  by  three  quantities 
taken  in  one  order  of  rotation,  it  is  usual  to  employ  the 
reciprocals  of  the  intercepts,  as  some  of  the  final  results 
are  simplified  by  this  means.  These  will  be  called  the 
three  parameters  of  the  plane,  and  will  be  denoted  by 
h,  k,  I  in  particular,  and  by  any  letter  or  quantity  in 
general. 

107.  Write  the  faces  of  the  cube  in  the  parametric  notation. 

108.  Show  that  (1,  1,  1),  or,  in  general  (a,  a,  a),  is  a  face  of 
the  regular  octahedron. 

109.  How  does  the  plane  (1,  1, 1)  cut  the  cube  ?  And  how  does 
(1,0,  0)  cut  the  octahedron  ? 

110.  A  plane  with  three  equal  parameters  trimcates  a  corner  of 
the  cube.  Describe  what  is  meant  by  truncating  a  comer  of  a  cube, 
and  compare  the  dihedral  angles  formed. 


234  SOLID   OK   SPATIAL   GEOMETRY. 

111.  A  plane  with  two  parameters  equal,  and  the  third  zero, 
truncates  an  edge  of  the  cube.  What  is  the  character  of  this 
truncation  ?  Compare  the  dihedral  angles  formed  ;  what  is  their 
value  ? 

112.  Show  that  the  plane  (1,  0,  0)  truncates  the  corner  of  the 
octahedron ;  and  that  (1, 1,  0)  truncates  an  edge  of  the  octahedron. 

113.  A  plane  with  three  unequal  parameters  bevels  a  corner  of 
the  cube.    Describe  the  operation  of  beveling  a  corner  of  the  cube. 

114.  The  plane  having  two  parameters  equal  and  the  third  un- 
equal, all  being  finite,  cuts  the  corner  of  the  cube  in  a  way  which 
will  be  called  trunco-bevelment.     Define  this  term. 

115.  By  what  change  does  a  trunco-beveling  plane  of  a  corner 
of  a  cube  become  a  truncating  plane  of  the  edge  ? 

116.  The  cube  admits  of  eight  truncating  planes  to  the  corners. 
Describe  the  figure  formed,  on  the  supposition  that  all  these  planes 
are  equidistant  from  the  centre,  and  the  faces  of  the  cube  are  com- 
pletely cut  away. 

117.  The  cube  admits  of  twelve  truncating  planes  to  the  edges. 
Describe  the  figure  formed  by  these  planes.  (This  is  the  rhombic 
dodecahedron.) 

118.  To  what  figure  does  the  plane  (1,  1,  0)  belong?  The 
plane  (1,  0,  1)  ? 

119.  How  does  the  plane  (a,  a,  0)  cut  the  octahedron  ? 

120.  The  cube  admits  of  two  beveling  planes  at  each  of  its  twelve 
edges.  Explain  how,  and  describe  the  figure  to  which  these  faces 
belong.     (This  is  the  tetra-hexahedron,  or  four-faced  cube.) 

121.  To  what  figure  do  the  planes  (1,  2,  0)  and  (2,  1,  0)  belong? 

122.  a.  The  cube  admits  of  three  trunco-beveling  planes  at  each 
corner.  How  many  faces  has  the  figure  to  which  these  planes 
belong  ? 

6.  Show  that  these  planes  may  be  disposed  in  two  different 
ways,  and  describe  the  difference  in  the  resulting  modification  of 
the  original  corner.  What  relation  does  it  hold  to  the  octahedron  ? 
(This  is  the  triakis-octahedron,  or  three-faced  octahedron.) 


MISCELLANEOUS    EXERCISES.  235 

123.  The  cube  admits  of  six  beveling  planes  at  each  corner. 
Write  these  planes,  and  give  the  character  of  the  figure  formed. 
(This  is  the  hexakis-octahedron,  or  six-faced  octahedron.) 

124.  To  what  figures  do  the  planes  (1,  1,  2),  (1,  1,  0),  (1,  2,  3), 
(1,  2,  2)  belong? 

Figures  formed  from  the  cube  by  putting  in  all  the 
possible  planes  given  by  varying  the  order  of  the  param- 
eters in  any  one  symbol,  as  (1,  2,  3),  (2,  3, 1),  (—  2, 1,  3), 
etc.,  are  called  holohedral  figures.  Those  formed  by 
putting  in  one-half  the  possible  planes,  in  alternate  posi- 
tions, are  hemihedral  figures. 

125.  One  beveling  plane  is  put  in  at  each  edge  of  a  cube  so  as  to 
alternate  the  positions  of  these  planes.  Show  that  the  resulting 
figure  will  have  pentagonal  faces.  (This  is  the  pentagonal  dodeca- 
hedron.) 

126.  Show  that  the  tetrahedron  is  a  hemihedral  form  derived 
from  the  cube,  and  give  its  mode  of  derivation. 

127.  The  cube  admits  of  three  beveling  planes  at  each  comer, 
applied  in  alternate  positions.  Write  these  planes  and  show  how 
they  are  applied.    (The  figure  is  the  pentagonal  icosi-tetrahedron.) 

128.  The  cube  admits  of  six  beveling  planes  at  four  corners  alter- 
nate in  position.  Write  these  planes.  (The  figure  is  the  hexa- 
tetrahedron. 

129.  If  p  be  the  length  of  normal,  from  the  origin,  on  the  plane, 
and  a,  /3,  7  be  the  direction  angles  of  p  (Art.  98),  show  that 
cos  a  =  hp,  cos  /3  =  kp,  and  cos  7  =  Ip. 


130.    Show  that p==l/VPT*2+T^. 


131.  Show  that  cos  o  =  /i/  y/K^  +  fc"^  +  V^,  with  symmetrical  ex- 
pressions for  cos  /3  and  cos  7. 

132.  If  6  be  the  angle  between  the  normals  to  two  planes, 
cos e=pp'{hh'  +  kk'+  W),  where  the  accented  letters  refer  to  the 
second  plane. 


236  SOLID   OK   SPATIAL  GEOMETRY. 

133.  Show  that 

cos  e  =  Qih'  +  kk'  +  ll')/y/{{h'^  +  k'^  +  1-)  (h''^  +  k''^  +  l'^)). 

134.  The  angle  between  the  planes  (1,  1,  1)  and  (1,  1,-1)  is 
cos~i|. 

135.  The  angle  between  the  planes  (1,  0,  0)  and  (1,  1,  1)  is 
cos~i^V3. 

136.  Find  the  cosine  of  a  dihedral  angle  of  the  regular  tetra- 
hedron. 

137.  Find  the  cosine  of  the  dihedral  angle  of  a  regular  octahe- 
dron. 

138.  Find  the  cosine  of  the  angle  between  a  face  of  the  cube 
and  that  of  the  octahedron. 

139.  The  type  plane  (1,  2,  3)  cuts  the  cube.  Find  the  angle 
between  two  adjacent  planes,  and  also  between  one  of  these  planes 
and  an  adjacent  face  of  the  cube. 

140.  Find  the  angle  between  (1,  2,  3;  and  (1,  1,  1). 

141.  The  edge  made  by  the  planes  (a,  b,  0)  and  (6,  —  a,  0)  is 
truncated  by  the  plane  (b  +  a,  b  —  a,  0). 

142.  Determine  the  ratios  of  the  intercepts  of  any  plane  which 
bevels  the  edge  of  the  rhombic  dodecahedron. 

143.  The  face  of  a  pentagonal  dodecahedron  being  (0,  1,  o)  with 
necessary  variations,  show  that  for  the  regular  figure  a—h(y/5±l), 
and  thence  show  that  the  cosine  of  a  dihedral  angle  of  this  figure 
is  iV5. 


INDEX  OF  TERMS,  ETC. 


The  nnmbers  refer  to  the  articles. 


ABT. 

Abscissa 207 

Acute  ppd 54 

Anchor  ring    .....  164 

Angle  of  obliquity   .     .     .  116 

Ant-orthogonal  projection  177 

Asymptotes 191 

Axes  of  space      ....  8 

Axial  pencil 5 

Axis  of  conic 194 

Basal  edges 50 

Capacity 106 

Centre  of  figure  ....  172 

Centroid 160 

Circle  of  contact      ...  86 

Col-unar  triangle     .     .     .  225 

Common  line 15 

Complanar 6 

Cone 67,  131 

Cone-circle 10 

Conic 189 

Conjugate  diameters    .     .201 

Conjugate  triangles      .     .  219 

Conspheric 83 

Co-ordinate  planes  ...  8 

Comer 33 


ABT. 

Cube 57 

Cuboid 57 

Cylinder 75,  136 

Cylindroid 162 

Developable  surface     .     .  165 

Diametral  plane  ....  78 

Diclinic  ppd 57 

Dihedral  angle    ....  23 

Direction  angles ....  98 

Direction  cosines     ...  98 

Direction  edges  ....  53 

Director 7 

Directrix 196 

Eccentricity 194 

Ellipse 190 

Equator 215 

Euler's  theorem ....  48 

Figure  of  revolution     .     .  149 

Focal  distance     ....  194 

Focus 194 

Frustum 59 

Generator 7 

Great  circle 77 

Groin 144 

Hyperbola 190 


237 


238 


INDEX. 


ART. 

Isoclinal 36 

Lamina 119 

Lateral  edges 50 

Lune 217 

Median 51 

Middle  section    ...    51,  128 

Monoclinic  ppd 57 

Nappes  of  cone   ....  67 

Net 65 

Normal 8 

Normal  plane      ....  8 

Oblique  prism      ....  60 

Obtuse  ppd 54 

Ordinate 194 

Orthogonal  or  orthographic 

proj 177 

Parabola 190 

Parallel  section  ....  20 

Parallelepiped     ....  53 

Perspective  projection .     .  177 

Planar 8 

Plane 1 

Plane  of  lines      ....  4 

Plane  section 19 

Point  circle     .....'  191 

Point  ellipse 191 

Polar  plane 86 

Polar  triangle      ....  226 

Pole 215 

Polyhedral  angle      ...  33 

Polyhedron 47 

Power  of  a  point      .     .     .  102 

Prism 60 

Prismatic  element  .     .     .  162 

Prismatoid 126 

Prismoid 126 


Prismoidal  formula . 
Projection  .... 
Pyramid     .... 
Quadrantal  triangle 
Eadical  centre     .     . 
Radical  line    .     .     . 
Radical  plane      .     . 
Reciprocal  corners  . 
Rectilinear  hyperbola 
Representative  corners 
Right  corner  .     .     . 
Right-bisector  plane 
Right-circular  cone 
Right  prism    .     . 
Right  section .     . 
Ruled  surface 
Secant  line     .     . 
Secant  plane  .     . 
Segment  of  sphere 
Semiaxis  major  . 
Semiaxis  minor  . 
Semivertical  angle 
Sheaf  of  lines  and  planes 
Sheets  of  a  cone  . 
Skew  quadrilateral 
Skew  surface  .     . 
Small  circle    .     . 
Solid  angle      .     . 
Solid  contents     . 
Sphere  .... 
Spheric  arc     .     . 
Spheric  figure 
Spheric  line    .     . 
Spheric  radius     . 
Spheric  triangle  . 
Spherical  excess . 


INDEX. 

239 

ART. 

ART. 

Surface  of  revolution 

.        .           66 

Unit  volume  .     .     . 

.     .     109 

Symmetrical  .     .     . 

.    39,  219 

Vanishing  line    .     . 

.     .     181 

Tangent  cone .     .     , 

.     .       86 

Vanishing  point  .     . 

.     .     181 

Tangent  sphere  .     . 

.     .       83 

Volume      .... 

.     .     106 

Triclinic  ppd.      .     . 

.     .       57 

Wedge 

.     .     124 

Trihedral  angle  .     . 

.     .       34 

Zone  of  sphere    .     . 

.     .     139 

B92 


oc 


-^.^^.x^oxt;  soxid  geometrj 

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